Irving  Stringham 


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NE  WCOMB'  S 

Mathematical  Course, 


I.    SCHOOL  COURSE. 

Algebra  for  Schools, $1.20 

Key  to  Algebra  for  Schools,        ....  1.20 

Plane  Geometry  and  Trigonometry,  with  Tables,  1.40 

The  Essentials  of  Trigonometry,   ....  1.25 

II.    COLLEGE   COURSE. 

Algebra  for  Colleges, $1.60 

Key  to  Algebra  for  Colleges,       ....  1.60 

Elements  of  Geometry, 1.50 

Plane  and  Spherical  Trigonometry,  with  Tables,  2.00 

Trigonometry  (separate), 1.50 

Tables  (separate), 1.40 

Elements  of  Analytic  Geometry,      .        .        .  1.50 

Calculus  (in  preparation), 

Astronomy  (Newcomb  and  Holden,)     .         .        .  2.50 

The  same,  Briefer  Course,          ....  1.40 


HENRY  HOLT  &  CO.,  Publishers.  New  York. 


NEWG0ME8  MATHEMATICAL   SERIES 


ELEMENTS 


OF 


ANALYTIC  GEOMETRY 


BY 

SIMON  NEWCOMB 

Professor  of  Mathematics,    U.  8.  Naiyy 


NEW  YORK 
HENRY  HOLT  AND  COMPANY 

1884 


Copyright,  1884, 

By 

HENRY  HOLT  AND   CO. 


PEEFACE. 


The  author  has  endeavored  so  to  arrange  the  present  work 
that  it  shall  be  adapted  both  to  those  who  do  and  those  who 
do  not  desire  to  make  a  special  study  of  advanced  mathema- 
tics. Believing  it  better  that  a  student  should  learn  a  little 
thoroughly  and  understandingly  than  that  he  should  go  over 
many  subjects  without  mastering  them,  the  work  is  so  con- 
structed as  to  offer  a  wide  range  of  choice  in  the  course  to  be 
selected. 

The  opening  chapter  contains  a  summary  of  the  new  ideas 
associated  with  the  use  of  algebraic  language,  which  the  stu- 
dent is  now  first  to  encounter.  His  subsequent  progress  will 
depend  very  largely  on  the  ease  and  thoroughness  with  which 
he  can  master  this  chapter. 

The  next  seven  chapters  correspond  closely  to  the  usual 
college  course  in  plane  analytic  geometry;  but  the  second  sec- 
tions of  Chapters  III.  and  IV. ,  as  well  as  some  sections  of  Chap- 
ter VIII. ,  may  be  regarded  as  extras  in  this  course. 

If  to  this  be  added  the  part  on  geometry  of  three  dimen- 
sions, we  shall  have  a  course  for  those  who  expect  to  apply  the 
subject  to  practical  problems  in  engineering  and  mechanics. 

The  second  sections  of  Chapters  III.  and  IV.,  together 
with  Part  III.,  form  an  introduction  to  the  modern  projective 
geometry;  a  subject  whose  elegance  especially  commends  it  to 
the  student  of  mathematical  taste.  The  author  has  tried  to 
develop  it  in  so  elementary  a  way  that  it  shall  offer  no  diffi- 
culty to  a  student  who  has  been  able  to  master  elementary 
geometry  and  trigonometry. 

800557 


TABLE    OF    CONTENTS. 


PART  1.     PLANE  ANALYTIC  GEOMETRY. 

Chapter  I.    Fundamental    Conceptions    m    Algebra    and 

Geometry  Page      3 

Algebraic  Conceptions,  3.  Roots  of  Quadratic  Equations,  7. 
Proportional  Quantities,  8.     Geometric  Conceptions,  11. 

Chapter  II.    Co-ordinates  and  Loci 13 

Cartesian  or  Bilinear  Co-ordinates,  13.  Problems,  15.  Area 
of  Triangle,  20.  Division  of  a  Finite  Line,  21.  Polar  Co-ordi- 
nates, 23.     Transformation  of  Co-ordinates,  25.    Loci,  29. 

Chapter  III.    The  Straight  Line 35 

Section  I.  Elementary  Theory.  Equation  of  a  Straight 
Line,  35.  General  Equation  of  the  First  Degree,  38.  Forms 
of  the  General  Equation,  39.  Special  Cases  of  Straight  Lines, 
42.  Problems,  43.  Normal  Form,  46.  Lines  Determined  by 
given  Conditions,  47.  Relation  of  Two  Lines,  51.  Transforma- 
tion to  New  Axis  of  Co-ordinates,  56. 

Section  II.  Use  of  the  Abbreviated  Notation.  Functions  of  the 
Co-ordinates,  58.     Theorems  of  the  Intersection  of  Lines,  62. 

Chapter  IV.    The  Circle 73 

Section  T.  Elementary  Theory.  Equation  of  the  Circle,  73. 
Intersection  of  Circles,  76.  Polar  Equation,  78.  Tangents  and 
Normal,  78.  Systems  of  Circles,  88.  Imaginary  Points  of  In- 
tersection, 91. 

Section  II.  Synthetic  Oeometry  of  the  Circle.  Poles  and 
Polars,  94.  Centres  of  Similitude,  98.  The  Radical  Axis,  103. 
Systems  of  Circles,  105.     Tangent  Circles,  106. 

Chapter  V.    The  Parabola 113 

Equation  of  the  Parabola,  113.  Polar  Equation,  115.  Diame- 
ters, 116.  Tangents  and  Normals,  117.  Equations  referred  to 
Diameter  and  Tangent,  123.     Poles  and  Polars,  125. 


Vi  TABLE  OF  CONTENTS. 

Chapter  VI.     The  Ellipse 131 

Equations  aud  Fundamental  Properties,  131.  Polar  Equa- 
tion, 135.  Diameters,  138.  Conjugate  Diameters,  140.  Equa- 
tion referred  to  Conjugate  Diameters,  144.  Supplemental 
Chords,  145.  Relation  of  the  Ellipse  and  Circle,  146.  Area  of 
the  Ellipse,  147.  Tangents  and  Normals,  148.  Reciprocal 
Polar  Relations,  157.     Focus  and  Directrix,  161. 

Chapter  VII.    The  Hyperbola 107 

Equation  and  Fundamental  Properties,  167.  Equilateral 
Hyperbola,  170.  Conjugate  Hyperbola,  173.  Polar  Equa- 
tion, 174.  Diameters,  177.  Conjugate  Diameters,  179.  Equa- 
tion referred  to  Conjugate  Diameters,  183.  Tangents  and 
Normals,  184.  Poles  and  Polars,  189.  Focus  and  Directrix, 
•      191.     Asymptotes,  192. 

Chapter  VIII.  General  Equation  of  the  Second  Degree.  201 
Fundamental  Properties,  201.  Change  of  Direction  of  Axes, 
203.  Classification  of  Loci,  205.  The  Parabola,  206.  The 
Pair  of  Straight  Lines,  208.  Summary  of  Conclusions,  210. 
Similar  Conies,  212.  Families  of  Conies,  214.  Focus  and 
Directrix,  217. 


PART  II.     GEOMETRY  OF  THREE  DIMENSIONS. 

Chapter  I.    Position  and  Direction  in  Space 223 

Directions  and  Angles  in  Space,  223.  Projections  of  Lines, 
224.  Co-ordinate  Axes  and  Planes,  225.  Distance  and  Direc- 
tion between  Points,  230.  Direction-Cosines,  234.  Trans- 
formation of  Co-ordinates,  237.    Polar  Co-ordinates  in  Space,  240. 

Chapter  II.    The  Plane. ,... , 245 

Loci  of  Equations,  245.  Equation  of  the  Plane,  246.  Gen- 
eral Equation  of  the  First  Degree,  247.  Relations  of  Two  or 
More  Planes,  254. 

Chapter  III.    The  Straight  Line  in  Space 261 

.♦  Equations  of  a  Straight  Line,  261.  Symmetrical  Equations, 
263.  Direction- V'ectors,  265.  Common  Perpendicular  to  Two 
Lines,  267.     Intersection  of  Line  and  Plane,  271. 

Chapter  IV.    Quadric  Surfaces 275 

General  Properties  of  Quadrics,  275.  Centre  and  Diameter, 
377.     Conjugate  Axes  and  Planes,  279.     Diametral  Planes,  280. 


TABLE  OF  CONTENTS.  Vll 

Principal  Axes,  282.  The  Three  Classes  of  Quadrics,  283.  The 
Ellipsoid,  283.  The  Hyperboloid  of  One  Nappe,  284.  The 
Hyperboloid  of  Two  Nappes,  286.  Tangent  Lines  and  Planes, 
287.  Generating  Lines  of  the  Hyperboloid  of  One  Nappe,  289. 
Poles  and  Polar  Planes,  294.  Special  Forms  of  Quadrics,  298. 
The  Paraboloid,  298.  The  Cone,  299.  The  Pair  of  Planes, 
300.     Surfaces  of  Revolution,  300. 

PART  III.     INTRODUCTION  TO  MODERN  GEOMETRY. . .  305 

The  Principle  of  Duality,  305.  The  Distance-Ratio,  308. 
The  Sine-Ratio,  311.  Theorems  involving  Distance-  and  Sine- 
Ratios,  316.  The  Anharmonic  Ratio,  321.  Permutation  of 
Points,  323.  Anharmonic  Ratio  of  a  Pencil  of  Lines,  326.  An- 
harmonic Properties,  327.  Projective  Properties  of  Figures,  332. 
Harmonic  Points  and  Pencils,  334.  Anharmonic  Properties  of 
Conies,  338.  Pascal's  Theorem  and  its  Correlative,  341. 
Trilinear  Co-ordinates,  343.     Line-Coordinates,  348. 


ANALYTIC    GEOMETRY. 


PART    I. 
PLANE    ANALYTIC    GEOMETRY. 


CHAPTER  I. 


FUNDAMENTAL  CONCEPTIONS    IN   ALGEBRA  AND 
GEOMETRY. 


1.  Analytic  Geometry  is  a  branch  of  mathematics  in  which 
position  is  defined  by  means  of  algebraic  quantities. 

As  an  example  of  how  position  may  be  defined  by  quanti- 
ties we  may  take  latitude  and  longitude.     The  statement 

"This  ship  is  in  lat.  47°  N.  and  long.  52°  W.'' 
indicates  to  the  expert  a  certain  definite  point  on  the  earth's 
surface  near  Newfoundland. 

47°  and  52°  are  the  quantities  indicating  the  position. 

Algebraic  Conceptions. 

The  following  algebraic  conceptions  and  principles  should 
be  well  understood  by  the  student  of  Analytic  Geometry. 

2.  Principles  of  Algehraic  Language. 

I.  When  an  algebraic  symbol  is  used  in  a  statement,  the 
statement  is  considered  true  for  any  value  of  that  symbol, 
unless  some  limitation  is  placed  upon  it. 

II.  Any  algebraic  expression  represents  a  quantity^  and 
may  itself  be  represented  by  a  single  letter. 


4  PLANE  ANALYTIC  GEOMETRY. 

3.  Constants  and  Variables.     A  quantity  is  called 
Constant  when  a  definite  fixed  value  is  supposed  to  be 

assigned  it; 

Variable  when,  no  definite  value  being  assigned  it,  it  is 
subject  to  change. 

We  may,  when  we  please,  assign  a  definite  value  to  a 
variable.     It  then  becomes,  for  the  time  being,  a  constant. 

Example.     We  may  regard  the  expression 
x^  —  a^x 

as  one  in  which  x  may  take  all  possible  values,  while  a  remains 
constant:  x  is  then  a  variable. 

But  we  may  also  inquire  what  definite  value  x  must  have 
in  order  that  the  expression  may  vanish.  We  readily  find 
these  values  to  be 

a;  =  0;        x  —  a\        x  =  —  a. 

The  quantity  x  then  becomes  a  constant. 

Again,  we  may  think  of  a  constant  as  undergoing  variation: 
it  then  becomes  a  variable. 

Remark.  The  distinction  of  constant  from  variable  is 
not  an  absolute  but  only  a  relative  one;  that  is,  relative  to 
otlier  quantities,  or  to  our  way  of  thinking  at  the  moment. 
The  only  absolute  constants  are  arithmetical  numbers. 

4.  Functions.  When  two  variables  are  so  related  that  a 
change  in  one  produces  a  change  in  the  other,  the  latter  is 
called  a  function  of  the  other. 

Such  relations  between  quantities  are  expressed  by  alge- 
braic equations. 

Example.     In  the  equation 

y  =  ax^h, 
a  change  in  a,  5  or  x  produces  a  change  in  y.     Hence  y  is  a 
function  of  these  quantities. 

The  value  of  an  algebraic  expression  containing  any  sym- 
bol will  generally  vary  with  that  symbol.     Hence 

Any  algehraic  expression  containing  a  variable  is  a  func- 
tion of  that  variable. 


FUNDAMENTAL   CONCEPTIONS.  5 

IndeiKiident  Variables.  When  a  quantity,  y,  is  a  func- 
tion of  another  quantity,  x,  we  may  assign  to  x  all  possible 
values,  and  study  the  corresponding  values  of  y. 

The  quantity  x  is  then  called  an  independent  variable. 

5.  Identical  and  Conditional  Equatio7is.  An  equation 
between  algebraic  symbols  may  be  either 

necessarily  true,  whatever  values  be  assigned  the  symbols; 

or  true  only  when  some  relation  exists  between  those 
values. 

An  equation  necessarily  true  is  called  an  identical  equa- 
tion, or  an  identity.  An  equation  only  conditionally  true 
is  called  an  equation  of  condition,  or  a  condition. 

Between  the  two  members  of  an  identity  the  sign  =  is 
used;  between  those  of  a  condition,  the  sign  =. 

Example.    We  have 

{x  -\-  a)  {x  —  a)  -\-  a^^  x^; 

because  the  two  members  are  necessarily  equal  for  all  values 
of  X  and  a.     But  the  statement 

ax  —  by  =  0 

can  be  true  only  when 

b 

a  ^' 
and  is  therefore  a  condition. 

The  question  whether  an  equation  is  an  identity  or  a  con- 
dition is  settled  by  reducing  or  solving  it. 

If  an  identity,  the  two  members  may  be  reduced  to  the 
same  expressions,  or,  if  we  try  to  solve  it,  we  shall  only  bring 
out  0  =  0. 

If  a  condition,  a  value  of  x  in  terms  of  the  remaining 
quantities  will  be  possible. 

Theorem.  An  equation  of  condition  becomes  an  identity 
by  solving  it  with  respect  to  any  one  symbol,  and  suhstituting 
the  value  of  the  symbol  thus  found  in  the  equation. 

Example.     If  in  the  ijreceding  equation 
ax  —  by  =  0 


6  PLANE  ANALYTIC  GEOMETRY. 

we  substitute  the  value  of  x  derived  from  it,  we  have 

a-y-  by  =  0, 

an  identity. 

Hence  any  eq^iatmi  may  he  regarded  as  an  identity  hy 
siqyj^osing  any  one  of  its  symbols  to  represent  that  function  of 
its  other  quantities  obtained  by  solving  it. 
Example.     The  equation 

aP  -by  =  (^ 
changes  into  the  identity 

aP-by  =  0 

when  we  suppose  P  ^—y, 

6.  The  symbol  =  is  also  used  as  the  symbol  of  definition 
when,  in  accordance  with  §  2,  II.,  we  use  a  symbol  to  repre- 
sent an  expression.     For  example, 

ax  -{-  by  ~  X 
means, 

*^  we  use  Xfor  brevity,  to  represent  the  expression  ax  -\-  hyJ^ 
When  the  sign  =  follows  an  expression  in  this  way,  it  may 
be  read,  '^  which  let  us  caliy 

7.  Lemma.  Between  the  variables  x  arid  y  and  the  con- 
stants A,  B  and  C  the  identity 

Ax-]-  By  -{-  C=Q  (1) 

subsists  when,  and  only  vjhen, 

^  =  0,         ^  =  0,         C  =  0.  (2) 

Proof.  That  the  identity  subsists  in  the  case  supposed 
is  obvious;  that  it  subsists  only  in  this  case  is  seen  by  showing, 
first,  that  if  G  were  different  from  zero,  the  identity  would 
fail  for  a;  =  0,  ?/  =  0;  and  next,  that,  C  being  zero,  the 
identity  would  fail  for  a:  =  0  when  B  was  finite,  and  for 
y  :=  0  when  A  was  finite. 

Kemark  1.     The  deduction  of  (2)  from  (1)  rests  on  the 
assumption  that  x  and  y  are  independent  variables.     If  A  and 
B  were  regarded  as  variables,  the  conclusions  would  be 
X  =  0\        y  =  0;         (7=0. 


FUNDAMENTAL   CONCEPTIONS.  7 

Eemark  2.  Note  the  great  difference  between  the  inter- 
pretation of  the  equation 

ax  -\-  hy  -{-  c  =  0 

and  of  the  identity 

ax  -\-  hy  -\-  c  =  0. 

The  equation  expresses  a  certain  relation  between  the  vari- 
ables X  and  y  such  that  to  each  definite  value  of  x  corresponds 
a  definite  value  of  y^  namely,  the  value 

ax  +  c 

y  =  — r- 

The  identity  expresses  no  relation  between  the  quantities, 
but  requires  zero  values  of  a,  h  and  c, 

8.  Roots  of  Quadratic  Equations.  Every  quadratic  equa- 
tion is  considered  to  have  two  roots,  which  may  \)Q  real  and 
unequaly  real  and  equal,  or  imaginary.     If  the  equation  is 

ax^  -f  J^  -f  c  =  0,  (a) 

then,  since  we  know  the  roots  to  be  given  by  the  equation 


-I  ±^V  -  4:ac  2c 

X  =  — 


2a  _  ^>  ip  |/J2  _  ^ac 

we  see  that  the  roots  will  be 

real  when  b^  —  ^ac  >  0,  i.e.,  when  h^  —  4ac  is  positive; 

real  and  equal  when  H^  —  ^ac  =  0; 

imaginary  when  V^  —  4ac  <  0;  i.e.,  when  Z>^  —  ^ac  is 
negative. 

The  student  should  now  be  able  to  explain  the  following 
special  cases: 

1.  If  the  absolute  term  c  vanishes,  the  roots  become 

X  =  0        and        x  = . 

a 

2.  If  J  and  c  both  vanish,  both  roots  become  zero. 

3.  If  a  approaches  zero  as  its  limit,  one  root  increases 

without  limit,  and  the  other  approaches  the  limit   —  j. 


8  PLANE  ANALYTIC  GEOMETRY. 

Hence  we  may  say:  When  the  coefficient  of  x^  in  the  quad- 
ratic equation  (a)  vanishes,  the  two  roots  are 

a:  =  —  T         and         a:  =  oo . 

0 

4.  If  both  a  and  b  vanish  while  c  remains  finite,  both 
roots  increase  to  infinity. 

5.  If  «,  h  and  c  all  vanish,  the  roots  are  entirely  indeter- 
minate, and  the  equation  is  satisfied  by  all  values  of  x, 

9.  Froportio7ial  Quantities.  The  quantities  of  one  series, 
a,  h,  c,  etc.,  are  said  to  be  proportional  to  those  of  another 
series.  A,  B,  C,  etc.,  when  each  quantity  of  the  one  series  is 
equal  to  the  corresponding  quantity  of  the  other  multiplied 
by  the  same  factor. 

The  fact  of  such  proportionality  is  expressed  in  the  vari- 
ous forms: 

a  :  A  =  b  :  B  —  c  :  G  =  etc.; 

a  :h  :  c  '.  etc.  =  A  :  B  :  C  :  etc.; 

a  =  pA,     b  =  pB,     c  =  pC,    etc.; 
p  being,  in  the  last  case,  the  multiplying  factor. 

TEST   EXERCISES* 

1.  A  point  at  the  distance  (1)  from  one  side  of  a  right 
angle  and  at  the  distance  (2)  from  the  other  side  will  be  at 
the  distance  (3)  from  the  vertex  of  the  angle. 

Here  the  student  will  substitute  symbols  at  pleasure  for 
(1)  and  (2),  and  will  replace  (3)  by  the  proper  function  of 
those  symbols,  reading  the  statement  accordingly.  For  ex- 
ample, he  may  put 

X  in  place  of  (1)  and  y  in  place  of  (2), 

y  -{-  xm  place  of  (1)  and  y  —  x'ln  place  of  (2), 
etc.  etc.  etc., 

*  These  exercises  are  designed  to  decide  the  question  whether  the 
student  has  a  suflScieut  command  of  algebraic  language  and  of  geometri- 
cal conceptions  to  enable  him  to  proceed  with  advantage  to  the  study 
of  Analytic  Geometry.  If  he  can  perform  all  the  exercises  with  ease, 
he  is  probably  well  prepared  to  go  on ;  if  he  performs  them  only  with 
diflflculty,  he  may  need  much  assistance  in  understanding  the  subject. 


FUNDAMENTAL   (JONGEPTIONIS.  9 

and  in  each  case  he  must  read  the  statement  witli  the  j^ropcr 
expression  in  place  of  (3). 

2.  If,  in  the  preceding  example,  (1)  varies  while  (2)  re- 
mains constant,  the  point  will  describe  a  line  to 

Fill  the  blanks  with  appropriate  words. 

3.  If  (2)  varies  while  (1)  remains  constant,  the  point  will 
describe  a  line to  the . 

4.  If  (1)  remains  equal  to  (2),  but  both  vary,  tlie  point 
will  move  along  the . 

5.  If  (1)  and  (2)  vary  in  such  a  way  that  (3)  remains  con- 
stant, the  point  will  describe  a  of  radius  around 

as  a  — . 

6.  If  two  fixed  points,  A  and  B,  are  at  the  distances  from 
each  other,  and  if  a  third  point,  P,  be  taken  at  the  distance 
(4)  from  each  of  these  points;  then,  if  (4)  varies,  the  point  P 
will  describe  a  line  {define  the  situation  of  tlie  Ime).  But,  in 
varying,  the  distance  (4)  cannot  become  less  than  — . 

The  numbers  in  parentheses  are  to  be  replaced  by  appro- 
priate symbols  or  expressions. 

7.  If,  in  the  preceding  example,  a  point  be  taken  at  the 
distance  (5)  from  the  point  ^4,  and  at  the  distance  (6)  from  B-, 
then,  if  (5)  varies  while  (6)  remains  constant,  the  point  will 
describe  a of around as  a  — . 

8.  But  if  (6)  varies  while  (5)  remains  constant,  the  point 
will 

9.  If  the  constant  value  of  (6)  in  Ex.  7  plus  the  con- 
stant of  (5)  in  Ex.  8  =  a,  the  two will  be to  each 

other. 

10.  If  a  line  be  drawn  so  as  to  pass  at  the  distance  r 
from  each  of  the  preceding  points,  then,  if  r  varies,  the 
line  will  turn. round  {describe  liow  it  will  turn).      But  the 

value  of  r  can  never  exceed  ,  and  for  each  value  of  r 

there  will  be  two  positions  of  the  line  making  equal  angles 
with  — . 

11.  If  the  line  be  required  to  pass  at  the  distance  (7)  from 
the  point  A,  and  at  the  distance  (8)  from  the  point  B\  then, 
if  (7)  varies  while  (8)  remains  constant,  the  line  v/ill  move 


10  PLANE  ANALYTIC  OEOMETRT. 

round  so  as  always  to  be  tangent  to  the around  —  as 

a with  radius  — . 

12.  What  symbols  must  (9)  and  (10)  be  replaced  by  in 
order  that  all  values  of  x  and  y  which  satisfy  the  equation 

Ax-\-  By  -\-  G  =0 

may  also  satisfy  the  equation 

mAx  +  (9)?/  +  (10)  =  0; 

that  is,  in  order  that  these  two  equations  may  give  the  same 
value  of  y  in  terms  of  a;? 

13.  Show  that  the  identity 

ax  -\-  hy  -\-  c  ^  Ax  -^  By  -\-  G, 
X  and  y  being  variables,  is  impossible  unless 
a  =A;        b  =  B;        c  =  G. 

14.  If  we  put 

F  =  x  -  2y  -}-3c,        P'  =  3x  -  6y  +  9c, 
is  it  possible  to  form  an  identity  of  the  form 
P  +  mP'  =  0, 

and,  if  so,  what  will  be  the  value  of  ?n? 

15.  Generalize  the  preceding  result  by  showing  that  if  we 
have 

f  P  =  ax-\-by  -j-  c,        P'  =  Ax  -^  By  ~\-  G, 

the  identity  P  +  mP'  =  0, 

X  and  y  being  variables,  is  possible  only  when 

a  \  A  =  l  :  B  =  c  I  G, 
and  express  the  value  of  m, 

16.  11  a  -\-  X  remains  constant,  and  x  varies  at  the  rate  of 
plus  one  foot  a  second,  at  what  rate  will  a  vary? 

17.  What  will  be  the  answer  to  this  last  example  if  it  is 
(I  —  2x  instead  of  a  -\-  x  which  remains  constant? 

18.  If  X  may  take  any  values  between  the  extremes  —  1 

and  +  2,  between  what  extremes  will  the  value  of  — '—z,  be 

X  —  1 

contained? 


FUNDAMENTAL  CONCEPTIONS.  H 

Geometric  Conceptions. 

10.  A  geometric  concept,  form,  or  figure  of  any  kind,  may 
be  called  a  geometric  object.* 

In  the  higher  geometry  all  geometric  objects,  when  not 
qualified,  are  considered  as  complete  in  every  particular. 

Examples.  A  straight  line  is  considered  to  extend  to 
infinity  in  both  directions.  When  a  terminating  straight  line 
is  treated,  it  is  considered  as  that  portion  of  an  infinite  straight 
line  contained  between  some  two  points. 

A  triangle  is  considered  as  formed  by  three  indefinite 
straight  lines  intersecting  each  other  in  three  different  points. 

Geometric  objects  differ  from  each  other  in  magnitude, 
form  and  situation. 

Points,  straight  lines  and  planes  can,  however,  differ  only 
in  situation,  because  any  two  points,  any  two  lines  or  any  two 
planes  may  be  made  to  coincide  with  each  other  by  a  change 
of  situation. 

11.  Points  at  Infinity.  A  pair  of  parallel  lines  are  said  to 
intersect  in  ?i  point  at  infinity;  that  is,  in  a  point  at  an  infinite 
distance. 

The  idea  of  a  point  at  infinity  is  reached  in  this  way:  Let 
us  suppose  one  of  two  intersecting  lines  to  turn  round  on  one 
of  its  points  and  gradually  to  approach  the  position  of  parallel- 
ism to  the  other  line.  As  this  position  is  approached  the 
point  of  intersection  of  the  two  lines  will  recede  indefinitely,  in 
such  wise  that  while  the  revolving  line  approaches  parallelism 
as  its  limit,  the  point  will  recede  beyond  every  assignable  limit. 

Conversely,  if  we  suppose  the  point  of  intersection  to  recede 
indefinitely  along  the  fixed  line,  the  moving  line  will  approach 

*  This  is  the  best  English  word  which  has  presented  itself  to  the 
author  to  correspond  to  the  Gebild  of  the  Germans.  Such  a  word  is  needed 
in  the  higher  geometry  as  a  term  of  the  most  general  kind  to  express 
the  things  reasoned  about.  The  term  magnitude  is  too  limited,  not  only 
because  a  point  is  to  be  included  among  geometric  objects,  but  because 
objects  are  considered  not  merely  as  magnitudes  but,  in  a  more  general 
way,  as  things  of  which  magnitude  is  only  one  of  the  qualities. 


12  PLANE  ANALYTIC  GEOMETRY. 

the  position  of  parallelism  as  its  limit.  This  limit  will  be 
the  same  whether  the  point  of  intersection  recedes  in  one 
direction  or  in  the  opposite. 

Hence,  using  the  convenient  language  of  infinity,  we  see 
that  when  the  point  of  intersection  is  at  infinity  in  either 
direction  the  two  lines  are  parallel.  There  is,  therefore,  no 
need  of  making  any  distinction  between  these  supposed  points, 
and  they  are  talked  about  as  a  single  point,  called  the  point  at 
infinity. 

The  principle  here  involved  is  of  extensive  application  in 
the  higher  mathematics,  and  may  be  expressed  thus:    , 

Instead  of  ^ismg  new  or  different  forms  of  language  to 
meet  excej^tional  cases,  we  use  the  common  language,  hut  put  an 
exce])tional  interpretatioyi  ujjon  it. 

The  advantage  of  this  way  of  speaking  is  that  we  are  not  obliged  to 
make  any  exceptional  cases  respecting  the  intersection  of  lines  when  the 
two  lines  become  parallel. 

The  proposition,  Two  straight  lines  intersect  in  a  single  point,  is  then 
considered  universally  true,  the  point  being  at  infinity  when  the  lines 
are  parallel. 

The  following  is  a  convenvient  illustration  of  this  form  of  language. 
Let  it  be  required  to  draw  a  line  through  a  fixed  point,  P,  so  as  to  inter- 
sect   the   fixed    line  h  at    the  ^ 

same  point,  Q,  where  the  line         ~h ^„^--^^ 

a  intersects  it.  The  construc- 
tion will  be  literally  possible  so 
long  as  a  and  b  intersect,  but 
will  cease  to  be  literally  possible  ^ 
if  a  takes  the  position  a',  paral- 
lel to  b,  because  then  there  will  be  no  point  Q  of  intersection. 

But  let  us  interpret  the  problem  in  this  way:  The  required  line  must 
intersect  b  where  a  intersects  b.  But  in  case  of  parallelism,  a  intersects 
b  nowhere.  Hence  the  required  line  must  intersect  b  nowhere;  that  is, 
it  must  be  parallel  to  it.  It  is  this  particular  noiohere  which  is  called  the 
point  at  infinity  on  the  line  b.  It  is,  moreover,  clear  that  if  Q  recedes  to 
infinity,  both  the  line  a  and  the  required  line  will  approach  the  position 
of  parallelism  to  b  as  their  respective  limits. 


CHAPTER  II. 

OF   CO-ORDINATES  AND    LOCI.       ' 

12.  Def.  The  co-ordinates  of  a  geometric  object  are 
those  quantities  whicli  determine  its  situation. 

Co-ordinates,  like  other  quantities,  are  represented  by 
numerical  or  algebraic  symbols. 

The  situation  of  an  object  is  defined  by  its  relations 
to  some  system  of  points  or  lines  supposed  to  be  fixed.  Such 
a  system  is  called  a  system  of  co-ordinates. 

There  are  several  systems  of  co-ordinates  to  be  separately 
defined. 

First  System :    Cartesian  or  Bilinear 
Co-ordinates. 

13.  On  this  system  the  position  of  a  point  is  fixed  by  its 
relation  to  two  intersecting  straight  lines  called  axes. 

Let  AX  and  BY  \iQ  the  two  lines,  and  0  their  point  of 
intersection.  -y- 

The  point    0  is  then  / 

called  the  origin.  / 

The     indefinite     line  y- yP 

AX,  which  we  may  con-  / 

ceive  to  be  horizontal,  is  / 

called  the  axis  of  ab-  A  '^ ~ 

scissas,  or  the  axis  of  X.  / 

The  intersecting  line  / 

BY'i^  called  the  axis  of  / 

ordinates,  or  the  axis  ^' 

of  r. 

Let  P  be  the  point  whose  position  is  to  be  defined. 


14  PLANE  ANALYTIC  GEOMETRY. 

From  P  draw  PM  parallel  to  OY  and  meeting  the  axis 
of  X  in  Mj  and 

PiV  parallel  to  OX  and                                  ^ 
meeting  the  axis  of  J^in                                  / 
N.  n/ ,p 

Then    either    of    the  / 

equal  lines   OM,   NP  is  / 

called  the  abscissa  of  the  -^ J. Z X 

point  P;  I  J&. 

Either    of    the    equal  / 

lines  MP,  ON  is   called  / 

the    ordinate    of  ■  the  -g/ 

point  P. 

It  is  evident  that  for  every  position  we  assign  to  P  the 
abscissa  and  ordinate  will  each  have  a  definite  value. 

14.  Co-ordinates  Detennine  a  Point.  When  the  lengths 
OM  and  MP  are  given,  the  point  P  is  completely  determined 
in  the  following  way:  We  measure  from  0  on  the  axis  of  X 
the  given  distance  OM. 

Through  M  we  draw  an  indefinite  line  parallel  to  the  axis 
of  Y,  and  on  this  line  measure  a  length  MP. 

The  single  point  P  which  we  thus  reach  is  the  point  which 
has  the  given  abscissa  and  ordinate. 

Because  the  abscissa  and  ordinate  thus  determine  the 
situation  of  P,  they  form,  by  definition,  a  pair  of  co-ordi7iates 
ofP(§12). 

Notation.     The  abscissa  is  represented  by  the  symbol  x. 
The  ordinate  is  represented  by  the  symbol  y. 

It  is  evident  that  if  the  point  P  be  fixed  in  position,  its 
co-ordinates  will  be  constants.  But  if  P  varies,  one  or  both 
of  the  co-ordinates  will  vary  also. 

15.  Algelraic  Signs  of  the  Co-ordinates.  In  what  pre- 
cedes it  is  supposed  that  the  direction,  as  well  as  the  distance, 
of  the  measures  OM  and  MP  is  given.  If  these  directions 
were  arbitrary,  we  might  measure  the  given  distance  OM  in 
either  direction  from  0,  and  thus  reach  either  the  point  Mio 
the  right  of  0  or  the  point  M'  to  the  left  of  0. 


CO-ORDINATES  AND  LOCI. 


15 


By  measuring  the  ordinate  in  either  direction  from  the 
points  M  and  M'  we  sliould  reach  cither  of  four  points, 
P,  P',  P",  jP'",  of  which  the  co-ordinates  would  all  be  equal 
iw  absolute  value. 

To  avoid  ambiguity  in  this  respect  the  algehraic  sign 
of  the  abscissa  is  supposed 

positive  when  meas- 
ured from  0  towards  the 
right,  and 

negative  when  meas- 
ured towards  the  left. 

The  ordinate  MP  is 
supposed 

positive  when  meas- 
ured upiDard,  and 

negative  when  meas- 
ured doivnward. 

Now  if  the  abscissa 
X  =  OM  =  a  and  the 
ordinate  y  =  MP  =  d,  then  the 

co-ordinates  of  P  are  x  =  -[-  a, 
co-ordinates  of  P'  are  x  =  —  a, 
co-ordinates  of  P"  are  x  =  ^  a, 
co-ordinates  of  P'"  are  x  =^  -\-  a. 


y  =  +  ^; 
y  =  -  h 
y  =  —  b. 


Thus  the  ambiguity  is  completely  avoided  when  the  algebraic 
signs  of  the  co-ordinates,  as  well  as  their  absolute  values,  are 
given,  so  that  only  one  point  corresponds  to  one  pair  of  alge- 
braic values  of  the  co-ordinates. 

16.  Rectangular  Co-ordinates.  When  not  otherwise  ex- 
pressed, the  axes  of  co-ordinates  are  supposed  to  intersect  at 
right  angles.  The  co-ordinates  are  then  called  rectangular 
co-ordinates. 

To  designate  a  point  by  its  co-ordinates  we  enclose  the 
symbols  or  numbers  expressing  the  co-ordinates  between  pa- 
rentheses, with  a  comma  between  them,  writing  the  value  of 
X  first. 


16  PLANE  ANALYTIC  GEOMETRY. 

Example.  By  (2,  3)  we  mean  ^*the  point  of  which  the 
abscissa  is  2  and  the  ordinate  is  3." 

EXERCISES. 

1.  Draw  a  pair  of  rectangular  axes,  and,  taking  a  centi- 
metre or  inch,  as  may  be  most  convenient,  for  ihe  unit,  lay 
down  the  position  of  points  having  the  following  co-ordinates: 

(+  2,  +  3),  (+  2,  -  3),  (-  2,  +  3),  (-  2,  -  3), 
(+  3,  +  2),  (+  3,  -  2),  (-  3,  +  2),  (-  3,  -  2). 

Show  that  these  eight  points  all  lie  on  a  circle  having  the 
centre  as  its  origin.     What  is  the  radius  of  this  circle? 

2.  Mark  a  number  of  points  of  each  of  which  the  ordinate 
shall  be  equal  to  the  abscissa.     How  are  these  points  situated? 

3.  Mark  the  points  (1,  -  1),  (2,  -  2),  (-  1,  1)  and 
(—  2,  2),  and  show  their  relations. 

4.  Mark  the  points  (1,  2),  (2,  4),  (3,  G),  (4,  8),  and  show 
how  they  are  situated  relatively  to  each  other. 

5.  If  we  join  the  points  {a,  —  i)  and  {a,  h)  by  a  straight 
line,  what  will  be  the  direction  of  this  line? 

6.  Find,  in  the  same  wa}-,  the  direction  of  the  line  joining 
the  points  {a,  h)  and  (—  a,  b);  (a,  h)  and  {—  a,  —  h). 

7.  Show  that  the  distance  of  the  point  {a,  h)  from  the 
origin  is  Va""  -f  W. 

8.  If  we  mark  all  possible  points  for  which  y  has  the  con- 
stant value  -f- 1,  how  will  these  points  be  situated? 

1*7.  Problem  I.  To  express  the  distaiice  between  two 
^joints  ivhose  co-ordinates  are  giveyi. 

When  the  co-ordinates  of  two  points  are  given,  the  position 
of  each  point  is  completely  determined  (§  14). 

Therefore  the  distance  between  the  points  is  completely 
determined,  and  may  be  measured  geometrically. 

The  algebraic  problem  requires  us  to  express  this  distance 
algebraically  in  terms  of  those  quantities  which  determine  the 
position  of  the  points,  namely,  their  co-ordinates. 

In  the  figure  let  P'  and  Pbe  the  two  points;  x',  y\  the  co- 
ordinates of  P';  and  x,  y,  the  co-ordinates  of  F. 


CO-ORDINATES  AND  LOCI. 


17 


Then  we  shall  have 


OM'  =  x\ 
P'M'  ==  if, 


OM  =  x\ 
PM  =  y. 


If  from  P'  we  drop 
a  perpendicular,  P'R, 
upon  MP,  we  shall  have, 
from  the  right-angled 
triangle  P'PR,  ^ 

P'R  =  M'M  =x  - 

and  RP  =  MP  -  MR  =  y  - 

Then,  by  the  Pythagorean  proposition, 


Y 

p/ 

} 

/ 

R 

0 

b/ 

/ 

X 

/ 

j\:    M 

P'P'  =  P^R'  +  RP\ 

Let  us  then  put  d  =  the  distance  P'P. 
By  substituting  the  values  in  terms  of  the  co-ordinates  and 
extracting  the  square  root,  we  shall  have 

d  =  ^/{{x  -  x'r  +  {,j  -  y'y\,  (1) 

which  is  the  required  expression  for  the  distance  of  the  points 
in  terms  of  their  co-ordinates. 

IS.  Peoblem  II.  To  express  the  ajigle  which  the  line 
joining  two  points,  given  hy  their  co-ordinates,  malces  with 
the  axis  of  X, 

Using  the  same  construction  as  before,  let  B  be  the  point 
in  which  the  line  PP'  intersects  the  axis  of  X. 

The  required  angle  will  then  be 

PBX    or    PP'R, 

If  we  put 

■  f  E  the  required  angle, 

we  shall  have,  by  trigonometry, 

RP  —  P'P  sin  6  =  f7  sin  f; 

P'R  =  P'P  cos  e  z=,d  cos  f ; 
whence,  by  division, 

RP       y  -  ?/' 


tan  e 


P'R 


X  —   X' 


(3) 


18  PLANE  ANALYTIC  OEOMETRT. 

The  last  equation  gives  tlie  required  expression  for  the 
tangent,  from  which  e  may  be  found. 

19.  The  two  preceding  problems  may  be  more  elegantly 
solved  by  a  single  pair  of  equations: 


d  sin  €  —  y  — 
d  cos  e 


ii~y';,)  (3) 

X  —  X  .  ) 


The  method  of  solving  these  equations  is  explained  in 
trigonometry. 

20.  Problem  III.  Ttvo  points  heing  given  hy  their  co- 
ordinates, it  is  required  to  find  the  p)oints  in  ivhich  the  straight 
line  joining  them  intersects  the  respective  axes  of  co-ordinates. 

Solution.  Let  B  be  the  point  in  which  the  line  inter- 
sects the  axis  of  x,  C  the  point  in  which  it  intersects  the  axis 
oiy. 

The  point  B  will  then  be  given  by  the  value  of  OB,  its 
abscissa,  which  we  denote  by  x^,  and  C  by  the  value  of  00, 
its  ordinate,  which  we  denote  by  y^. 

In  the  similar  triangles  MBP  and  RP'P  we  have 

BM :  MP  =  P'R  :  RP. 

Substituting  for  the  lines  their  values  in  terms  of  the  co- 
ordinates, this  gives 

y  -y 

whence 

OB  =  OM  -  BM^x-  ^i^Zli'i 

y  -  y 

or         X  =  ^^^'  ~  ^'^  ~  y^^  "  ^'"^  =  ^y'  ~  ^''^  u\ 

y  -y'  y' -y  '      _ 

The  value  oi  00  can  be  found  by  a  similar  construction, 
but  we  may  also  deduce  it  from  OB  by  the  equation 

00  =  OB  tan  e. 

But  in  the 'figure  as  drawn  0  falls  below  0,  so  that  the 
value  oi  00  just  obtained  is  the  negative  of  the  required 


C0-0BDINATE8  AND  LOCI.  19 

ordinate  of  the  point  of  intersection.     This  co-ordijiate  being 
y^f  we  shall  have 

y,=  -OB  tan  «  =  ^?^X  (5) 


The  student  should  now  note  the  relation  between  the 
conditions  of  the  geometric  and  the  algebraic  solutions. 
The  problem  considered  as  a  geometric  one  is: 

Tivo  points  leing  given  in  position^  to  find  the  intersection 
of  the  straight  line  joining  them  luith  the  axes  of  co-ordi7iates» 

The  problem  is  solved  geometrically  simply  by  drawing  the 
line.     The  algebraic  requirement  is: 

Ttvo  points  being  giveii  hy  rneaiis  of  their  co-ordinates,  it  is 
required  to  express  the  poitiis  in  which  the  straight  line  join- 
ing  them  intersects  the  co-ordinate  axes  in  terms  of  the  respec- 
tive co-ordi7iates  of  the  given  points. 

The  algebraic  solution  is  given  by  the  equations  (4)  and 
(5). 

21.  The  preceding  problems  illustrate  the  following  gen- 
eral principle: 

Whenever  one  geometric  object  is  determined  by  another 
geometric  object,  the  algebraic  quantities  which  define  the  one 
can  be  exjjressed  in  terms  of  those  quantities  ivhich  define  the 
other. 

EXERCISES. 

1.  Lay  down  the  four  points  (1,  1),  (1,  2),  (2,  2),  (2,  1), 
and  join  each  one  and  that  next  following  so  as  to  form  a 
quadrilateral.     What  will  be  the  nature  of  this  quadrilateral? 

2.  Show  that  the  points  (1,  0),  (1,  1),  (2,  0),  (2,  1)  lie 
at  the  four  vertices  of  a  square. 

3.  Show  that  each  of  the  following  sets  of  four  points 
are  the  vertices  of  a  parallelogram : 

Set  (a):  (0,  0),     (3,  1),     (0,  4),     (3,  -  3)- 
'     Set  {b):  (1,  3),     (2,  5),     (6,  5),     (5,  3); 
Set  (c):  (1,1),     (2,4),     (5,5),     (4,2). 

4.  Show  by  a  geometric  construction,  employing  the 
properties  of  similar  triangles,  that  each  of  the  lines  joining 


PLANE  ANALYTIC  GEOMETRY. 


the  following  pairs  of  points  passes  through  the  origin  of  co- 
ordinates: 

(a):  a  line  joining  points  (1,  1)  and  (2,  2); 

(b):  a  line  joining  points  (1,  2)  and  (3,  6); 

(c) :  a  line  joining  points  (1,  3)  and  (—  1,-3); 

(d):  a  line  joining  points  (a,  b)  and  (71a,  nb). 

Show  in  the  same  way  that  each  of  the  following  trip- 


(3,  3); 
(3,  4); 
(1>  +  4); 
(-  1,  0); 

(a  -j-  npf  b  -\-  nq). 


0. 
lets  of  points  lies  in  a  straight  line: 

(a):  (1,1),  (2,2), 

{b):  (1,0),  (2,2), 

(6'):  (-1,0),  (0,  +2), 
(.0:(3,  -2),  (1,-1), 
(c):  {a,  b),  {a-\-p,b  -\-  q), 

6.  What  are  the  distance  and  direction  (relatively  to  the 
axis  of  X)  from  the  point  (1,  2)  to  the  point  (4,  G)? 

23,  Problem  IV.  To  find  the  area  of  a  triangle,  the 
co-ordinates  of  the  vertices  beiiig  given, 

Kemakk.  Since  the  positions  of  the  vertices  completely 
determine  the  triangle,  and  therefore  determine  its  area  also, 
it  follows  from  the  general  principle,  §  21,  that  this  area  can 
be  algebraically  expressed  in  terms  of  the  co-ordinates  of  the 
vertices. 

Solution.  Let  P,  P'  and  P"  be 
the  vertices,  and  {x,  y),  {x%  y')  and 
{x",  y")  their  respective  co-ordi- 
nates. 

Let  us  put  A  the  area  of  the 
triangle.     We  shall  then  have 

A  —  area  PMM"P^'  plus  area 
M'P'P"M''  minus  area  MWP'P. 

In  these  three  trapezoids  we  have 
area  MPP"M"    =  i{3IP  +  iWP'')MiW 

area  J/"P"P'J/'  =4(i¥"P"  +  M'P')M"M' 

area    MPP'M'    =  i(M'P'  +  MP)MM' 


0     M 


M"     M' 


-  .-^0; 


=  W  +  y)  (^'  -  ^). 


CO-ORDINATES  AND  LOCI. 


21 


Therefore 
2J  =  (2/  +  y")  (X" 


X)  +  iy"  +  y')  (^'  - 
+  (y'- 


x") 

[-  y)  (*■  -  ^'), 


')  +  y"{x'  -  X),        (6) 


or,  by  reduction, 

'ZJ  =  y(x"  -  x')  +  y\x  ■ 
which  is  the  required  expression. 

33.  To  divide  a  finite  line  into  segments  having  a  given 
ratio.  A  finite  line  is  defined  by  the  co-ordinates  of  its  two 
terminal  points.     Let  us  now  consider  the  problem: 

To  find  the  co-ordinates  of  the  poiiit  luhich  divides  the  finite 
line  joining  two  given  poifits  into  seginents  having  a  given 
ratio. 

Let  us  put: 

x^y  y^,  the  co-ordinates  of  one  end,  A,  of  the  line. 
x^,  y^,  the  co-ordinates  of  the  other  end,  B. 
A,  ^,  the  given  ratio. 

X,  y,  the  co-ordinates  of  the  required  point,  P. 
Draw  ^iVland  PQ  each 
parallel  to  the  axis  of  X,  and 
PM,  BN  each  parallel  to 
the  axis  of  Y.  Then 
AM  =  x  —  x^;  PQ  =  x^—x; 
PM=y-y^;  BQ  =  y^-y. 
Since  we  require  that 

AP  :  PB  =  ?i:  M, 
we  have  the  proportion 

X:  /.i  =  AP  :  PB 

=  AM:  PQ  =  X  -  X 
=  PM:BQ  =  y-y^ 
We  hence  deduce  the  equations 

X{x^  -  x)  =  /j{x  -  .tJ, 

Hy.  -  y)  =  M{y  -  y,), 

which  give 

\x,  -f  fix^^         ^_  _  Ay,  +  fAy 


:  x^ 

-y^ 


—  X 

-y- 


X  = 


y  = 


A  +  yU    '         ^  A^-/^ 

which  are  the  required  co-ordinates  of  the  point  of  division. 


(7) 


22  PLANE  ANALYTIC  OEOMETBT. 

Corollary.  If  P  is  to  be  the  middle  point  of  the  line,  we 
have  A  =  yw,  whence 

or. 

Each  co-ordinate  of  the  iniddle  point  of  a  line  is  half  the 
sum  of  the  co7Tes2J07idi7i(/  co-ordinates  of  its  terminal  poi?its. 

EXERCISES. 

1.  Express  the  co-ordinates  of  the  middle  point  of  the  line 
terminatiDg  in  the  points  (1,  6)  and  (3,  —  4). 

2.  One  end  of  a  line  is  at  the  point  {—2,  —  3)  and  its 
middle  point  at  (1,  —  2).     Where  is  the  other  end? 

3.  Find  the  middle  point  of  that  segment  of  the  line  join- 
ing the  points  (—  1,  6)  and  (3,  —  2)  which  is  contained  be- 
tween the  axes  of  co-ordinates.     A^is.  (1,  2). 

4.  A  line  terminating  at  the  points  (1,  6)  and  (3,  —  4)  is 
to  be  divided  into  four  equal  segments.  Find  the  co-ordi- 
nates of  the  three  dividing  points. 

5.  The  line  joining  the  points  {a,  h)  and  {p,  q)  is  to  be 
divided  mio  five  equal  parts.  Express  the  co-ordinates  of  the 
four  points  of  division. 

6.  What  is  the  distance  between  the  middle  points  of  the 
lines  whose  respective  termini  are  in  the  points  (1,  7),  (—  5,  3) 
and  (0,  2),  (6,  -  4)? 

7.  What  point  bisects  the  line  from  the  origin  to  the 
middle  point  of  the  line  terminating  at  the  points  (7,  —  0) 
and  (-  3,-7)? 

8.  Find  the  co-ordinates  of  the  point  which  is  two  thirds 
of  the  way  from  the  point  {a,  h)  to  the  point  {a' ,  h'). 

9.  Prove  the  theorem  that  the  three  medial  lines  of  a  tri- 
angle meet  in  a  point  two  thirds  of  the  way  from  each  vertex 
to  the  opposite  side,  as  follows: 

Let  {x^,  yj,  {x^,  y^)  and  {x^,  y^)  be  the  three  vertices  of 
the  triangle. 

Express  the  middle  point  of  each  side. 

Then  express  the  co-ordinates  of  those  three  points  which 


CO-ORDINATES  AND  LOCI.  23 

arc  respectively  two  thirds  of  the  way  from  the  several  vertices 
to  the  middle  points  of  the  opposite  sides,  and  thus  show  that 
the  three  points  are  coincident. 

10.  Prove  that  the  lines  joining  the  middle  points  of  the 
opposite  sides  of  a  quadrilateral  and  the  line  joining  the 
middle  points  of  the  diagonals  all  bisect  each  other. 

To  do  this,  express  the  co-ordinates  of  the  middle  points  of  the  sides 
and  of  the  diagonals,  and  then  of  the  middle  points  of  the  three  joining 
lines,  and  show  that  the  latter  points  are  the  same  for  each  joining  line. 
The  very  simple  proof  of  this  theorem  which  is  thus  found  affords  a 
striking  example  of  the  power  of  the  analytic  method. 

Second  System :   Polar  Co-ordinates. 

The  position  of  a  point  may  be  defined  by  its  distance  and 
direction  from  a  fixed  point. 

The  fixed  point  is  then  called  the  origin. 

The  distance  of  the  point  from  the  origin  is  called  the 
radius  vector  of  the  point. 

In  plane  geometry  the  direction  of  a  point  from  the  origin 
is  fixed  by  the  angle  which  the  radius  vector  makes  with  an 
adopted  base-line. 


Let  OX  be  the  base-line,  and  P  the  point;  P  being  in 
any  one  of  the  positions  P',  P",  etc. 

OP  will  then  be  the  radius  vector,  and  the  angle  XOP 
will  be  the  required  angle. 


24 


PLANE  ANALYTIC  GEOMETRY. 


We  generally  put 

r  =  the  radius  vector  OP,  and 

6  E  the  angle  XOP,  which  is  called  the  vectorial  angle. 

The  former  is  always  considered  positive,  being  measured 
from  the  origin,  0,  in  the  direction  OP.  The  latter  is  posi- 
tive when  measured  in  the  direction  opposite  to  that  in  which 
the  hands  of  a  watch  move,  and  negative  in  the  opposite  di- 
rection, just  as  in  trigonometry. 

34.  Problem.  To  express  the  distance  hekvee^i  two  points 
in  terms  of  their  polar  co-ordinates. 

Let  P  and  Q  be  the  points. 

In  the  triangle  P0§  we  have,    y 
by  trigonometry, 

P§==  r^  +  r"  -  2rr'  cos  POQ 
=zr''-\-r''-2rr'co^{d-e'), 

6  and  6'  being  the  angles  which 
the  radii  vectores  make  with  the 
initial  or  base  line;  therefore 


PQ  =  {r'  +  r''  -  %rr'  cos  {0 
which  is  the  distance  required. 


0')!' 


(9) 


EXERCISES. 

1.  Show  how  a  point  will  be  situated  when  its  vectorial 
angle  is  in  the  first,  second,  third  and  fourth  quadrant  re- 
spectively. 

2.  If  the  vectorial  angle  d'  and  radii  vectores  r  and  r*  are 
constant,  while  Q  may  vary  at  pleasure,  for  what  values  of  d 
will  the  distance  of  the  points  be  the  greatest  and  least  i)Ossi- 
ble,  and  what  will  be  the  greatest  and  least  distances?  Show 
the  correspondence  of  the  algebraic  answer  from  equation  (9) 
with  the  obvious  answer  from  the  figure. 

3.  If  r  =  r'  and  6  ■\- B'  =  180°,  show  both  geometrically 
and  algebraically  that  distance  =  2r  cos  0. 

4.  If  ^  -  0'  =  90°  or  270°,  express  the  distance  of  the 
points  both  by  a  diagram  and  by  the  equation  (0). 


CO-ORDINATES  AND  LOCI. 


25 


Transformation    of    Co-ordinates 
System  to  Another. 


from    One 


The  general  problem  of  the  transformation  of  co-ordi- 
nates is  this: 

Given:  1.  The  co-ordinates  x  and  y  of  a  point  P  referred 
to  some  system  of  co-ordinates. 

Given:  2.  The  position  of  a  second  system  of  co-ordinates 
m  relation  to  the  other  syste?)i. 

Kequired:  To  express  the  co-ordinates  of  P  when  referred 
to  the  second  system, 

25.  Relation  of  Rectangular  and  Polar  Co-ordinates. 

Let  OX,  OF  be  the  rectangular  axes,  and  P  the  position 
of  any  point. 

1st.  We  shall  suppose  the  origin  to  be  taken  as  the  pole, 
and  the  axis  of  abscissas  as  the  base  or  initial  line;  then  we 
shall  evidently  have  Y 

X  =  r  cos  0 
and  y  =  r  sin  0. 

To  express  the  poiar  co-ordi- 
nates in  terms  of  the  rectangular   ^ 
co-ordinates,  we  have  from  the '  ^       M 

last  two  equations,  by  squaring  and  adding, 
r^  =  x^  -\-  y^,         or 

and,  by  division,  tan   Q  = 


=  V^Tl/% 


which  determine  r  and  6  when  x  and  y  are  given 

2d.  If  the  initial  or  base  line 
instead  of  coinciding  with  the 
axis  of  X  makes  an  angle  a  with 
it,  we  shall  evidently  have,  from 
the  figure, 

X  =  r  cos  {a  -f  6) 
and  y  =  r  sin  (a  -\-  6), 

whence      r  =  Vx""  -f  y""      and       tan  {a  -J-  6) 


0^ 


x 


26  PLANE  ANALYTIC  OEOMETBY. 


EXERCISES. 

1.  In  the  figure  of  §  24,  express  the  area  of  the  triangle 
OPQ  in  terms  of  r,  r'  and  6  -  6'. 

2.  If  r,  r'  and  r"  are  the  i^dii  vectores,  and  6,  6'  and  6" 
tlie  corresponding  angles  of  three  points,  which  we  shall  call 
P,  F' ,  P" ,  it  is  required  to  express  the  areas,  first,  of  the 
triangles  OPP',  OP'P"  and  OPP",  and  then  of  PP'P'\ 

3.  The  point  (3,  3)  is  the  centre  of  a  circle  of  radius  2,  in 
which  two  diameters,  each  making  angles  of  45°  with  the  axes, 
are  drawn.  Find  the  polar  co-ordinates  of  the  ends  of  these 
diameters. 

4.  The  point  (a,  b)  is  the  centre  of  a  circle  of  radius  P. 
From  the  centre  is  drawn  a  radius  making  an  angle  y  with 
the  axis  of  X.  Express  the  rectangular  co-ordinates  of  the 
end  of  this  radius. 

26.  Transformation  from  one  rectaiigular  system  to 
anotlier. 

Solution,     Let  us  first  suppose 
the  two  systems  of  co-ordinates  par-  N 
allel.     Also  suppose 
OXy   OY  the  axes  of  the  original 

system ; 
O'X',  O'Y'  the  axes  of  the  second 

system; 
P  the  point  whose  co-ordinates  are  x  and  y  in  the  old  system. 

Draw 

PM'MW  Y0\\  Y'O', 
PN'N\\XO\\X'0', 
and  put 

a  =  the  abscissa  of  the  new  origin,  0',  referred  to  the 
old  system; 

b  =  the  ordinate  of  0'; 

x^  =  the  abscissa  O'l/'  of  P  referred  to  the  new  system; 

y'  =  the  ordinate  M^P  of  P  referred  to  the  new  system. 
We  then  have 


0  M 


M x' 

_X 


x'  -  X  -  «; )  ^^^^ 

y  =  y  -  h^ 


CO-ORDINATES  AND  LOCI. 


27 


which  are  the  required  expressions  for  the  new  co-ordinates  in 
terms  of  the  old  ones. 

3*7.  Secondly.     Suppose  the  new  axes  to  make  an  angle, 
6,  with  the  old  ones,  but  to  have  the  same  origin. 

Let  us  put 
r  E  the  radius  vector  0P\ 
q)  E  the  angle  XOP. 

We  shall  then  have 


Angle  X'OP  =  cp  -  6, 

Putting,  as  before, 
x'  and  y'  for  the  co-ordinates  referred  to  the  new  system,  and 
X  and  y  the  co-ordinates  of  the  old  system,  we  have,  by  §  25, 

X   =  r  cos  cp',  2/    =  ^'  sin  cp',  (a) 

x'  =  r  cos((p  —  S);        y'  =  r  s'm((p  —  6).  (d) 

By  trigonometry, 

cos((p  ■—  6)  =  cos  q)  cos  d  -|-  pin  q)  sin  S; 
sin(<^  —  6)  =  sin  cp  cos  S  —  cos  cp  sin  d. 

Substituting  these  values  in  (b)  and  eliminating  r  and  cp 
by  («),  we  have 

x^  =  y  sin  6  -}-  x  cos  d; )  ,. .  >. 

y'  =  y  cos  6  —  X  sin  d;) 

which  are  the  required  expressions. 

To  express  the  old  co-ordinates  in  terms  of  the  new  co- 
ordinates, we  have 

X  =  X*  cos  S  —  y^  sin  d; 
y  =  a;'  sin  (J  -f-  y'  cos 

If  we  take  for  (^  the  angle  which  the  new  axis  of  Y  makes 
with  the  old  axis  of  X,  the  new  axis  of  Xwill  make  an  angle 
oi  d  —  90°  with  the  old  one.  Hence  in  this  case  the  formulae 
of  transformation  will  be  found  by  writing  6  —  90°  for  d  in 
(12),  which  gives 


8;\ 

s.S 


(13) 


X  =  x'  sin  S  -\-  y*  cos  d\       ) 


y  —  —  o:^  cos  S  -\-  y'  sin  6. 


(13) 


28 


PLANE  ANALYTIC  OEOMETBY. 


28.  Thirdly.  Let  the  new  system  of  co-ordinates  have 
any  origin  and  direction  whatever,  and  let  us  put,  as  before, 

«,  J  E  the  co-ordinates  of  the  new  origin  referred  to  the 
old  system; 

(J,  the  angle  which  each  axis  of  the  new  system  forms 
with  the  corresponding  axis  of  the  old  one. 

Imagine  through  the  new  origin  0'  an  intermediate  system 
of  co-ordinates  parallel  to  the  old  system,  and  let  us  put  x^  and 
y,  the  co-ordinates  of  P  referred  to  this  intermediate  system. 

Then,  by  (10), 

X^  ^^  X  —  a\  y^  =  y  —  If, 


By  (11), 


Whence 


x'  =  y^  sin  d  -\-  x^  cos  d; 
y'  =  y^  COS  S  —  x^  sin  d. 


=  {y  —  h)  sin  S  -\-  {x  —  a)  cos  6-,  \ 
=z  (y  —  h)  cos  6  —  {x  —  a)  sin  d;  ) 


(14) 


which  are  the  required  expressions. 

29.   Transformation  from  rectangular  to  oblique  co-ordi- 
nates, the  origin  remaining  the  same. 

Let  OXy  OYhe  the  rectangu- 
lar axes,  and  0X%  OY'  the  ob- 
lique axes;  the  angle  XOX^  =  a, 
XOY'  =  /?;  and  let  x,  y  be  the 
co-ordinates  of  any  point  P  re- 
ferred to  the  rectangular  axes, 
and  x^,  y'  the  co-ordinates  of  the 
same  point  referred  to  the  oblique 
axes.     Then 

x=  0M=  0N-\-  M'Q 
=  OM'  cos  XOX  +  PM'  cos  XO  Y' 

{since  XOr=PM'Q) 

=  x'  cos  oc  -\-  y'  cos  ^, 


and 


y  =  PM  =  M'N-{-  PQ 
=  OM'  sin  XOX'  -f  PM'  sin  XOY' 
=  x'  sin  a  -\-  y'  sin  /J; 


C0-0BDINATE8  AND  LOCI.  20 

which  are  the  expressions  of  the  rcctaiigiihir  co-ordinates  in 
terms  of  the  oblique  ones.  If  we  express  the  oblique  co-or- 
dinates in  terms  of  the  rectangular  ones,  we  shall  have 

,       X  s\n  /3  —  7/  cos  /3  .  ,       y  cos  a  —  x  sin  a 

x'  = ■    ,,.    ^         '-  and         1/  =  ^ r-7-5- r — . 

sin(/i  —  a)  ■'  sm(P  —  a) 


Of  Loci. 

30.  The  first  fundamental  principle  of  Analytic  Geom- 
etry, as  developed  in  what  precedes,  may  be  expressed  thus: 

Having  chosen  a  system  of  co-ordinates,  then 

To  every  pair  of  values  of  the  co-ordinates  corresponds  one 
definite  point  in  the  plane. 

Let  us  now  suppose  that,  instead  of  the  co-ordinates  beiug 
given,  only  an  equation  of  condition  between  them  is  given. 
Then  we  may  assign  any  value  we  please  to  one  co-ordinate, 
and  find  a  corresj^onding  value  of  the  other.  To  every  such 
pair  of  corresponding  values  will  correspond  a  definite  point. 
Since  these  pairs  of  values  may  be  as  numerous  as  we  please, 
we  conclude: 

A  pair  of  co-ordinates  subjected  to  a  single  equation  of 
condition  may  belong  to  a  series  of  points  imlimited  in  num- 
ber. 

If  one  co-ordinate  varies  continuously  and  uniformly,  the 
other  will  vary  according  to  some  regular  law.  From  this 
follows: 

The  poi7its  whose  co-ordinates  satisfy  an  equation  of  co7i- 
dition  all  lie  on  one  or  more  lines,  straight  or  curved. 

Def  A  line,  or  system  of  lines,  the  co-ordinates  of  every 
point  of  which  satisfy  an  equation  of  condition  is  called  the 
locus  of  that  equation. 

3 1 .  Problem.     To  draw  the  locus  of  a  given  equation. 
Solution.     1.  By  means  of  the  equation  express  one  co- 
ordinate, no  matter  which,  in  terms  of  the  second. 

2.  Assign  to  this  second  co-ordinate  a  series  of  values,  at 
pleasure,  differing  not  much  from  each  other. 


30 


PLANE  ANALYTIC  GEOMETRY. 


3.  Find  each  corresponding  value  of  the  other  co-ordinate. 

4.  Lay  down  the  point  corresponding  to  each  pair  of  values 
thus  found,  and  join  all  the  points  by  a  continuous  line. 

5.  This  line  will  be  the  required  locus. 
Example  1.     Construct  the  locus  of  the  equation 

lOy  =  x'  -  X  -  ^. 

Assigning  to  x  values  from  —  10  to  +  10,  differing  by 
two  units,  we  have 


a;  =  -  10    I 


8    1-6 
6.814-3.8 


+ 


-2    1      0    1+2    14-4     1+6    1+8    14- 10 
+    .21-    .41-    .21+    .81+2.61+5.21+    8. 


Laying  down  the  positions  of  these  eleven  points  corre- 
sponding to  these  pairs  of  co-ordinates,  we  find  them  to  be  as 
in  the  annexed  diagram. 


X 


M 


Example  2.     Construct  the  locus  of  the  equation 

(y  -  ^y  4-  (^  -  13)'  =  100. 

From  this  quadratic  equation  we  obtain  for  the  value  of  y, 
in  terms  of  x, 

y  =  5  ±  VIOO  -  (a;  -  12)'. 

The  following  conclusions  follow  from  this  equation: 
1.  For  every  value  we  assign  to  x  there  will  be  two  values 
of  y,  the  one  corresponding  to  the  positive,  the  other  to  the 
negative  value  of  the  sum.     To  form  the  locus  we  must  lay 
down  both  of  these  values. 


CO-ORDINATES  AND  LOCI, 


31 


2.  If  the  value  of  {x 


case  when 


a;  <  3  or  x  > 


•  12)"^  exceeds  100,  which  will  be  the 
22,  the  quantity  under  the  radical 


sign  will  be  negative,  and  the  value  of  y  will  be  imaginary. 
This  shows  that  there  is  no  value  of  ?/,  and  therefore  no  point 
of  the  curve,  except  when  x  is  contained  between  the  limits 

2  <  .T  <  22. 

We  now  find  the  following  sets  of  corresponding  values  of 
X  and  y: 


2 

3 

4         G 

8 

10 

12 

14 

16 

18 

20 

21 

23 

5.0 

9.4 

11.0     13.0 

14.2 

14.8 

15.0 

14.8 

14.2 

13.0 

11.0 

9.4 

5.0 

5.0 

0.6 

-1.0  -3.0 

-4.2 

-4.8 

-5.0 

-4.8 

-4.2 

-3.0 

-1.0 

0.6 

5.0 

Laying  down  these  points  upon  a  diagram,  we  shall  find 
them  to  fall  as  in  the  annexed  figure. 


EXERCISES. 

Construct  the  loci  of  the  following  equations  to  rectangu- 
lar co-ordinates: 

1.  y  =  Zx^  -  X  -10. 

2.  ^  =  sin  X. 

3.  y  =  cos  X. 

Note.  Iu  the  last  two  exercises  we  should,  in  rigor,  adopt  the  unit 
radius,  57°  18',  as  the  unit  of  x.  But  a  more  convenient  and  equally 
good  course  will  be  to  take  60°  as  the  unit,  and  let  it  correspond  to  one 
inch  on  the  paper.  Lay  off  a  scale  of  sixths  of  an  inch  on  the  axis  of  X, 
and  let  the  successive  points,  one  sixth  of  an  inch  apart,  be  0°,  10°,  20°, 


82  PLANE  ANALYTIC  GEOMETRY. 

30% 360°.     At  each  poiut  erect,  as  an  ordinate,  the  corie- 

spouding  value  of  the  natural  sine  or  cosine,  and  draw  the  curve  through 
the  extremities.     The  curve  is  called  the  curve  of  sines. 

We  need  not  stop  at  360°,  but  may  continue  on  indefinitely.  The 
curve  will  be  a  wave-line,  the  parts  of  which  continually  repeat  them- 
selves. 


4. 

y    =1  +  1. 

5. 

y      =^x  ^  1. 

6. 

X      =\y'  -  3. 

7. 

5ru    =  ?/'  -  by  - 

5. 

8. 

lOx  =  y'  -^  - 

10. 

9. 

f     =x\ 

10. 

y      =  tan  x. 

11. 

y      =  sec  X. 

Note.  The  object  of  the  above  exercises  is  to  give  the  student  a 
clear  practical  idea  of  the  relation  between  an  equation  and  its  locus. 
He  should  perform  as  many  of  them  as  are  necessary  for  this  purpose. 
It  is  in  theory  indifferent  what  scale  of  units  of  length  is  used,  but  in 
practice  a  scale  either  of  millimetres  or  tenths  of  an  inch  will  be  found 
most  convenient. 

32.  Intersections  of  Loci.  Consider  the  following  prob- 
lem: 

To  find  the  point  or  points  of  intersection  of  two  loci  given 
hy  their  equations. 

Solution.  Since  the  points  in  qnestion  are  common  to 
both  loci,  their  coordinates  must  satisfy  doth  equations. 
Hence  we  have  to  find  those  values  of  the  co-ordinates  which 
satisfy  both  equations.  This  is  done  by  solving  the  equations 
algebraically,  regarding  the  co-ordinates  as  unknown  quanti- 
ties. 

If  the  equations  are  each  of  the  first  degree,  there  will 
be  but  one  pair  of  values  of  the  co-ordinates,  and  therefore 
but  one  point  of  intersection. 

If  the  equations  are  one  or  both  of  the  second  or  any 
higher  degree,  there  may  be  several  roots,  in  which  case  there 
will  be  one  point  for  each  pair  of  roots.  The  curves  will  then 
have  several  points  of  intersection. 


CO-ORDINATES  AND  LOCI.  33 

If  the  roots  arc  one  or  both  imaginary,  the  loci  will  not 
intersect  at  all.  This  is  expressed  by  calling  the  points  of 
intersection  imaginary. 

Example.  To  find  the  point  in  which  the  loci  whose 
equations  are 

3^»  +  2a;'  =  164 
and  y   =  2x   —  3 

intersect  each  other. 

We  have  here  a  pair  of  simultaneous  equatiouSj  one  of 
which  is  a  quadratic.  Substituting  in  the  first  the  value  of  y 
from  the  second,  we  have  the  quadratic  equation  in  x, 

Qx'  -  12x  =  155. 

The  solution  of  this  equation  gives 

x  =  l±\/^---=  -\-6.1S    or     -4.18. 

The  corresponding  values  of  y  are 

y  =  9.3G     or     -  11.36. 

We  have  in  this  theory  a  correspondence  between  the 
moUlity  of  a  point  in  space  and  the  variahility  of  an  alge- 
braic quantity,  which  is  at  the  basis  of  Analytic  Geometry. 
That  is: 

To  the  unlwiited  variability  of  the  co-ordinates  x  and  y 
corresponds  the  mobility  of  a  point  to  all  parts  of  a  plane. 

To  the  limited  variaUlity  of  co-ordinates  subjected  to  one 
equation  of  condition  corresponds  the  limited  mohility  of  a 
point  confined  to  a  straight  or  curve  line,  but  at  liberty  to 
move  anywhere  along  that  line. 

To  the  co7istancy  of  co-ordinates  required  to  satisfy  two 
equations  corresponds  the  immohility  of  a  point  required  to 
be  on  two  lines  at  once,  that  is,  confined  to  the  intersections 
of  two  lines. 

It  must  always  be  understood  that  liberty  to  occupy  any 
one  of  several  points,  as  when  the  curves  have  several  points 
of  intersection,  is  not  mobility. 


34  PLANE  ANALYTIC  GEOMETRY. 

EXERCISES. 

Find  the  points  of  intersection  of  the  following  loci: 

1.  i^  +  f  =1    and    ^-^  =  2. 
a       b  ha 

2.  x'-{-i/  =  ^     and     1+1=1. 

3.  1+1=1     and    x'  +  y'=l. 

4.  y^-  =  4:ax    and    x  —  VSy  =  2. 
Do  the  following  loci  intersect? — 

5.  3rc'  -  ?/'    =  -  4    and  x'-{-y'  -2x  =  0. 

X         2    ^ 
iKf  "^  3  and     9^;^  +  2bi/  =  225. 

Note.  The  special  values  of  the  coordinates  found  from  the  above 
exercises  are  constants,  the  relation  of  which  to  the  variables  may  be 
explained  by  thinking  thus :  The  co-ordinates  are  affected  by  a  love  of 
liberty  which  prompts  them  to  take  all  possible  values  so  long  as  we, 
their  masters,  do  not  subject  them  to  any  condition. 

If  we  require  them  to  satisfy  an  equation,  they  obey  us,  but  exercise 
their  liberty  by  assuming  all  values  consistent  with  that  equation. 

If  we  require  them  also  to  satisfy  a  second  equation,  we  deprive  them 
of  all  liberty  of  variation,  and  chain  them  down  to  the  special  values 
which  satisfy  both  equations. 

Again,  if  we  put 

P  =  ax -\-  by  -\-  c, 

then,  so  long  as  we  require  the  co-ordinates  to  satisfy  the  equation 
P=  0,  P  retains  this  zero  value.  But  if  we  rub  out  the  =  0,  and  leave 
only  the  symbol  P  without  any  equation,  the  co-ordinates  instantly  re- 
sume their  liberty,  and,  by  varying,  make  P  take  all  values  whatever. 


CHAPTER  III. 

THE  STRAIGHT  LINE. 


Section  I.   Elementary  Theory  of  the  Straight 

Line. 


The  Equation  of  a  Straight  Line. 

33.  Problem.     To  find  the  equation  of  a  straight  line. 

In  order  that  the  equation  of  any  locus  may  be  found,  the 
locus  must  be  so  described  that  the  position  of  each  of  its  points 
can  be  determined.  Hence,  to  find  the  equation  of  a  straight 
line,  we  must  suppose  the  data  which  determine  the  situation 
of  the  line  to  be  given.  This  may  be  done  in  various  ways, 
of  which  the  following  are  examples: 

I.  A  line  is  completely  deter-  y 
mined  if  the  point  in  which  it 
intersects  the  axis  of  X,  and  the 
angle  which  it  makes  with  that 
axis,  are  given.  Let  us  then 
suppose  given : 

The  abscissa  OR  =  a  oi  the 
point  E  in  which  the  line  inter- 
sects the  axis  of  X; 

The  angle  e  which  the  line  makes  with  that  axis. 

To  find  the  equation,  let  P  be  any  point  whatever  on  the 
line.  From  P  drop  the  perpendicular  P3f  upon  the  axis  of 
X.     If  we  then  put 

X,  y,  the  co-ordinates  of  P, 
we  shall  have 
MP  =  y  =  EM  tan  s  =  {OM  -  OE)  tan  e  =  (x  -  a)  tan  8. 


36 


PLANE  ANALYTIC  GEOMETRY. 


Hence,  putting  m  ee  tan  £,  we  have 

y  —  7n{x  —  a).  (l) 

Because  P  may  be  any  point  Avhatever  on  the  line,  tins  equa- 
tion must  subsist  between  the  co-ordinates  of  every  point  of 
the  line;  it  is  therefore  the  equation  of  the  line. 

Def.  The  slope  of  a  line  is  the  tangent  of  the  angle  which 
it  forms  with  the  axis  of  abscissas. 

II.  Let  the  slope  in  of  the  line,  and  the  ordinate  h  of  the 
point  in  which  the  line   cuts  the     y 
axis  of  I',  be  given. 

Using  the  same  notation  as  be- 
fore, we  readily  find 

MP  =  X  tan  e  -{-  b, 
or  y  =  7nx  -\-  b\  (2) 

which  last  is  the  required  equation. 

III.  Let  the  points  A  and  B  in  which  the  line  intersects 
the  axes  of  co-ordinates  be  given.     Let  us  then  put 

a  =  the  abscissa  OA  of  the 
l^oint  A  in  which  the  line  inter- 
sects the  axis  of  X; 

b  =  the  ordinate  of  the  point 
B  in  which  it  intersects  the  axis 
of  F. 

Then,  if  P  be  any  point  on 
the  line,  the  similar  triangles 
BOA  and  P3IA  give  the  pro- 
portion 


or 


b  :  a  = 

:  FM :  MA  =  y 

iVe  hence  derive 

ay  =  b{a-  x); 

is, 

bx  -\-  ay  -—  ab, 

a^  b         ^' 

(3) 


Def.  The  lengths  OA  and  OB  from  the  origin  to  the 
points  in  which  the  line  cuts  the  co-ordinate  axes  are  called 
the  intercepts  of  the  line  upon  the  respective  axes. 


THE  STBAIQHT  LINE.  37 

EXERCISES. 

1.  Write  the  equations  of  lines  passing  through  the  origin 
and  making  the  angles  of  45°,  30°,  120°,  135°,  150°  and  e, 
respectively,  with  the  axis  of  X, 

2.  If  the  intercept  of  a  line  on  the  axis  of  X  is  a,  and  if 
€  is  the  angle  which  it  makes  with  that  axis,  express  its  inter- 
cept on  the  axis  of  Y, 

3.  Write  the  equation  of  the  line  whose  intercept  on  the 
axis  of  J^  =  5  and  which  makes  an  angle  of  30°  with  the 
axis  of  X. 

4.  Form  the  equation  of  the  line  whose  intercept  on  the 
axis  of  X  is  a  and  which  makes  an  angle  of  45°  with  that 
axis. 

5.  Show  geometrically  that  tlie  inverse  square  of  the  per- 
pendicular from  the  origin  upon  a  line  is  equal  to  the  sum  of 
the  inverse  squares  of  its  intercepts  on  the  axes. 

Note.     The  inverse  square  of  a  is  1  -4-  a'. 

6.  Express  the  tangents  of  the  angles  which  a  line  makes 
with  the  co-ordinate  axes  in  terms  of  its  intercepts  upon  tliose 
axes,  and  explain  the  algebraic  sign  of  the  tangent. 

7.  Two  lines  have  the  common  intercept  a  upon  the  axis 

of  X;  the  difference  of  their  slopes  is  unity;  and  the  sum  of 

their  intercepts  upon  the  axis  of  Y  is  c.     Find  the  separate 

intercepts  upon  ?/,  and  show  that  the  equations  of  the  two  lines 

are 

X  2?/  X  2?/  ^ 

-  H =  1         and         -  -\ ~—  =  1. 

a       c  —  a  a       c  -\-  a 

8.  What  is  the  relation  of  the  two  lines, 

y  =  mx  -{-  b 
and  y  =  mx  -{-  b  -\-  c? 

9.  What  are  the  relations  of  the  series  of  lines, 

y  =  mx, 
y  =  mx  -\-  by 
y  z=  mx  -\-  2b, 
y  =  mx  -f  3b, 
etc.        etc.  ? 


38  PLANE  ANALYTIC  GEOMETRY. 

10.  What  is  the  relation  of  the  two  lines, 
1/  =  b  -\-  mx 
and  y  -z^  h  —  mx"^ 

Especially  show  where  they  intersect,  the  relation  of  the 
angles  they  form  with  tlie  axis,  and  the  angle  they  form  with 
each  other  in  terms  of  ^  =  arc  tan  m. 

34.  The  eqnations  (1),  (2)  and  (3)  are  examples  of  the 
nnmerous  forms  which  the  equation  of  a  right  line  may 
assume.     "We  have  now  to  generalize  these  forms. 

Def.  An  equation  of  the  first  degree  between  two 
variables,  x  and  y,  means  any  equation  which  can  be  reduced 
to  the  form 

Ax-\-By^G=0,  (4) 

A,  B  and  C  being  any  constant  quantities  whatever. 

Theorem.  Every  eqication  of  the  first  degree  between  rect- 
angular co-ordinates  represents  a  straight  line. 

Proof.     The  equation  (4)  may  be  reduced  to  the  form 

.=  -^(.  +  5).  (5) 

Since  the  tangent  of  a  varying  angle  takes  all  values,  we 

can  always  find  an  angle,  =  s,  whose  tangent  shall  b& 


B' 


Q 

On  the  axis  of  X  measure  a  distance ^  =  a. 

A 


Then,  by  (1),  the  locus  of  (5)  will  be  the  line  which  inter- 
sects the  axis  of  J^  at  the  point  x  =  a  and  makes  an  angle  e 
with  the  axis  of  X.  Since  such  a  line  is  always  possible,  the 
theorem  is  proved. 

Scholium.  The  result  of  the  above  theorem  may  be  ex- 
pressed as  follows: 

The  locus  of  the  equation 

Ax-[-By-{-  C=0 
is  that  straight  line  which  intersects  the  axis  of  X  at  the 

C 

distance  —  -r  from  the  origin  and  makes  with  that  axis  an 

a7igle  whose  tangent  is  —  ^. 


THE  STRAIGHT  LINE.  39 

35.  Reductioji  of  the  General  Equation.  Any  pair  of 
values  of  x  and  y  which  satisfy  the  equation 

^a;  -f-  %  +  C  =  0 
must  also  make 

m{Ax  +  %  +  C)  =  0; 

that  is, 

{mA )  X  +  {7nB)y  -{-  wC  =  0. 

Hence,  since  m  may  be  any  quantity  whatever. 

If  we  multiiily  or  divide  all  the  coefficients,  A,  B  and  C, 
xohich  enter  into  the  general  equation,  hy  the  same  factor  or 
divisor,  the  line  represented  ly  the  equation  will  not  he  altered. 
Example.     The  equations 

y  —  2.1-  +  1  =  0, 
2y  —  4:3;  +  2  =  0, 
5?/  -  10a;  4-  5  =  0, 

all  represent  the  same  line,  because  they  all  give  the  same 
value  of  y  in  terms  of  x,  namely, 

y  =  2x  -  1. 

The  same  result  may  be  expressed  in  the  form : 

The  line  represented  ly  the  equation  (4)  depends  only  on 

the  mutual  ratios  of  the  coefficients  A,  B  and  C,  and  not 

upon  their  absolute  values.'^ 

36.  From  this  it  follows  that  special  forms  of  the  general 
equation  may  be  obtained  by  multiplying  or  dividing  it  by  any 
quantity. 

I.  First  Form.     By  dividing  by  B  we  obtain 

5^  +  y  +  S  =  o, 

or  AC  .^. 

*  This  introduction  of  more  quantities  than  are  really  necessary  for 
the  expression  of  a  result  is  quite  frequent  in  Mechanics  and  Geometry. 
It  has  the  advantage  of  enabling  us  to  assign  such  values  to  the  super- 
fluous quantities  as  will  reduce  the  expression  to  the  most  convenient 
form. 


40  PLANE  ANALYTIC  GEOMETRY. 

which  becomes  identical  with  tlie  form  (2)  by  putting 


™h-|;        iH-J                        (7) 

11.  Second  Form,     By  dividing  by  C  the  general  equation 
becomes 

^x  +  -^y  +  l=:d,                               (8) 

or 

A         B 

which  becomes  identical  witli  (3)  by  putting 

0         ,          0 
''--A'        ^^-B- 

III.   Third  or  Normal  Form.     Let  us  divide  by  VA^-\-  B\ 
The  equation  will  then  become 

ABC 

^  +  -:7^^=f=%^y  +  ^7^F^^=0.      (10) 


VA""  +  B'  VA'  -\-  B'  VA'^  b 

If  we  now  determine-  an  angle  a  by  the  equation 

B 


sin  a 


VA'  +  B' 
we  shall  have 

cos  a  =  Vl  —  silica  =  — .  (H) 

Va'  +  b'  ^   ^ 

Let  us  also  put,  for  brevity, 

O 


P  = 


VA'  +  B' 

The  general  equation  of  the  line  will  then  become 

a:  cos  a  +  y  sin  a  —  jt?  =  0,  (12) 

which  is  called  the  Normal  form  of  the  equation  of  a 
straight  line. 


THE  STRAIGHT  LINE.  41 


EXERCISES. 


Express  each  of  the  following  equations  in  the  forms  (2), 
(3)  and  (10): 

1.  3a;  +  4?/  +  15  =  0.  2.  4a:  -f-  3?/  -  15  =  0. 
3.  12x  -  5?/  -  13  =  0.  4.  X  -2y  -{-  6  =  0. 
5.       X  -{-    y  -\-   c  =  0.         6.     X  —    y  —    c  =  0. 

3  7 .  Relation  of  the  General  Equation  to  its  Special  Forms. 
The  forms  (1),  (2)  and  (3)  are  examples  of  numerous  special 
forms  under  which  the  equation  of  a  straight  line  may  be 
written.  The  general  form  is  not  to  be  regarded  as  a  distinct 
form,  but  as  a  form  which  may  be  made  to  express  all  others 
by  assigning  proper  values  to  the  constants  A,  B  and  0. 
For  example: 

The  form  (1)  is  equivalent  to 

y  —  mx  +  ma  =  0, 
which  is  what  the  general  form  becomes  when  we  put 

A  =  —  m, 
B=l, 
0  =  am. 
In  the  same  way,  to  reduce  the  general  form  to  (2),  we 

have  only  to  put 

A  =  —  7n, 

B=l, 

C=-h. 

To  reduce  it  to  (3)  we  put 

.    (7=1. 
Again,  the  normal  form  is  one  expressed  by  the  general 
form  when  we  suppose 

A  =  cos  a, 
B  =  sin  a, 


42 


PLANE  ANALYTIC  GEOMETRY. 


"We  may  also  say  that  the  normal  form  is  one  in  which 
A'  ^  B'  =  1. 

38.  Since  all  forms  of  the  equation  of  a  straight  line  are 
special  cases  of  the  general  form,  we  conclude: 

If  we  demonstrate  any  theorem  hy  means  of  the  general 
form  of  the  equatmi  of  a  straight  line,  that  demonstration  tuill 
include  all  the  special  forms. 

39.  Def.  The  constants  A,  B  and  C  which  enter  into 
the  equation  of  a  straight  line  are  called  its  parameters. 

The  parameters  determine  the  situation  of  a  line  as  co- 
ordinates do  the  position  of  a  point. 

Only  two  parameters  are  really  necessary  to  determine  the 
line,  but  there  is  often  a  convenience  in  using  three,  as  in  the 
general  form. 

A  line  is  completely  determined  when  its  parameters  are 
given.     Instead  of  saying, 

*'  The  line  whose  equation  \q  Ax  -\-  By  -\-  C  —  0," 
we  may  say, 

"The  line  (^1,  B,  (7)." 

40.  Special  Cases  of  Straight  Lines. 

I.  If,  in  the  general  equation  of  the  straight  line, 

Ax  -\-  By  -^  C  =  0, 

the  coeflBcients  A  and  B  are 
of  opposite  signs,  x  must  in- 
crease with  y,  the  line  makes 
an  acute  angle  with  the  axis  of 
X,  and  its  positive  direction  isQ' 
in  the  first  or  third  quadrant. 
QR  is  such  a  line. 

II.  If  A  and  B  are  of  the 
same  sign,  one  co-ordinate  diminishes  as  the  other  increases, 
the  line  makes  an  obtuse  angle  with  the  axis  of  X,  and  its 
positive  direction  is  in  the  second  or  fourth  quadrant. 

PS  is  such  a  line. 


THE  STRAIGHT  LINE.  43 

III.  If  A  vanishes,  the  equation  may  be  reduced  to 

y  = 7>  ~  ^  constant, 

while  X  may  have  any  value  whatever. 

The  line  is  then  parallel  to  the  axis  of  X  and  at  the  dis- 
tance —  -^  from  it. 

IV.  In  the  same  way,  the  equation  of  a  line  parallel  to 

the  axis  of  Y  is 

a;  =  a  constant, 

the  constant  being  the  distance  of  the  line  from  the  axis  of  Y. 

V.  If  this  constant  itself  vanishes,  the  line  will  coincide 
with  the  axis  of  Y.     Hence  the  equation  of  the  axis  of  y  is 

X  =  0. 

VI.  In  the  same  way,  the  equation  of  the  axis  of  x  is 

y  =  0. 


EXERCISES. 

1.  At  what  point  does  the  line 

ax  -\-  c  =  0 
cut  the  axis  of  X? 

2.  Write  the  equation  of  a  line  perpendicular  to  the  axis 
of  Xand  cutting  off  an  intercept,  d,  from  that  axis. 

3.  What  are  the  relations  of  the  four  lines, 

X  =  a-,        X  =  —  ct'y 

y  =  h      y  =  -h 

and  what  figure  do  they  form? 

41.  Special  Problems  co?mectecl  with  the  General  Equa- 
tion of  a  Straight  Line. 

I.  To  find  the  iiitercejjts  of  the  general  straight  line  iipon 
the  co-ordinate  axes. 

By  definition,  the  intercept  upon  the  axis  of  X  is  the  value 
of  X  when  y  =  0.  Putting  y  =  0  \n  the  general  equation,  it 
becomes 

Ax  +(7=0. 


44 


PLANE  ANALYTIC  OEOMETRT. 


a  = 


Hence,  if  we  put  a  for  the  intercept  upon  the  axis  of  X, 
we  have 

C 
A' 

In  the  same  way,  we  find  for  the  intercept  on  Y,  which  we 
call  b, 

II.   To  find  the  angle  which  a  line  makes  with  the  axis  of  X. 
We  have  already  shown  (§  34)  that 

A 
B' 

We  can  now  find  the  sine  and  cosine  of  e  by  trigonometric 
formulae,  as  follows: 

tan  e  A 


tan  e 


sm  £  = 


cos  €  = 


Vl  +  tan^f 
1 


VA'  +  B 
B 


(13) 


VI  +  tanV        VA""  +  B^ 

III.   To  ex2)ress  the  i^erpendicular  distance  of  a  'point  from 
a  given  line. 

Let  X*  and  y'  be  the  co-ordinates  of  the  point,  and 
Ax-^  By  -^  C  -^ 
the  equation  of  the  line. 

Since  the  position  of  the  point  is  completely  determined 
by  its  co-ordinates,  and  the  line 
by  its  parameters,  A,  B,  C,  the 
required  distance  admits  of  being 
expressed  in  terms  of  x',  y',  A, 
B  and  C. 

Let  P  be  the  point,  LN  the 
line,  and  PQ  the  perpendicular 
from  the  point  on  the  line;  and 
let  the  ordinate  PM  of  the  point 
intersect  the  line  in  11.  We  shall  . 
then  have  ^'^ 

PQ  ^  PE  cos  e.  (a) 


Y 

P 

K\ 

/ 

5>r 

"^ 

0 

/Q 

M. 

THE  STRAIGHT  LINE.  45 

Now  R  is  a  point  on  tlie  line  whose  abscissa  is  the  same  as 
that  of  P,  namely,  x']  and  if  we  put  RM  =  y^  =  the  ordi- 
nate of  R,  we  must  have,  since  R  is  on  the  line, 

Ax^  -{.  Bt/, -^  C  =  0, 

,  .  ,     .                                     Ax'  -i-  C 
which  gives  y^  = h • 

Then  PR  =  PM  -  RM  =  y'  -  y^ 

_B£_       _  Ax' -{.  By' -^r  O 
~   B         ^'  ~  B 

Substituting  in  {a)  this  value  of  PR  and  the  value  of  cos  e 

from  (13),  we  have 

^  VA'  -{-  B'  ^     ^ 

Since  the  co-ordinates  of  the  origin  are  x'  =  0  and  y'  =  0, 
we  have 

OQ'  =    ,     ^     --  (15) 

VA'  -{-  B'  ^     ^ 

which  gives  the  perpendicular  from  the  origin  on  the  line. 

EXERCISES. 

Find,  for  each  of  the  lines  represented  by  the  following 
equations, — 

The  angle  which  it  makes  with  the  axis  of  X; 

Its  intercepts  upon  the  axes; 

Its  distance  from  the  point  (4,  3); 

Its  least  distance  from  the  origin; 

The  length  of  that  portion,  intercepted  between  the  axes. 

1.  3a;  +    4?/  +  10  =  0.  2.  Sx  +  4?/  -  10  =  0. 

3.  6x  —  12y  4-  26  =  0.  4.     a;  +    ?/  =  0. 

5.  4:X  —    3y  —    5  =  0.  6.     x  —    y  =  0. 

X  11 

7.  X  COB,  a  -\-  ij  Bin  OL  —  p  =  0.     8.  ^ — h  t-     =  1. 

9.  Find  the  length  of  the  perpendicular  from  the  point 

X  II 

{a,  I)  on  the  line  —  -j-  ^  =  1,  and  show  that  it  is  equal  to 
the  negative  distance  of  the  line  from  the  origin. 


46 


PLANE  ANALYTIC  GEOMETRY. 


10.  Find  tlie  points  on  tlie  axis  of  X  which  are  at  a  per- 
pendicular  distance  a  from  the  line  — j-  •-  —  1  =  0. 

42.  Direct  Derivatioji  of  the  Normal  Form.     This  form 
may   be   derived    as   follows:      y 
From  the  origin  drop  the  per- 
pendicular OM  upon  the  line 
whose    equation   is   required. 


Let  P  be  any  point   of   the 
line,  and 

X  =  ON,  the  abscissa  of  P; 

y  =  NP,  its  ordinate; 

a  =  angle  NOMoi  the  per- 
pendicular with  the  axis  of  X\ 

P=  OM. 

From  N  draw  NQ  parallel  to  the  line,  and  PR  parallel  to 
OM.     Then 

OQ  =  OiV^cos  a  =  X  cos  a; 
QM  =  NP  sin  a  =  y  sin  a; 


Hence 


OQ  -\-  QM  =  p  —  X  cos  a  -\-  y  sin  a. 


X  cos  a  -\-  y  sin  a  —  p  —  0, 

which  is  the  normal  form  of  the  equation. 

We  hence  conclude: 

In  the  normal  form  the  parameters  p  and  a  arc  respectively 
the  perpendicular  from  the  origin  upon  the  line,  and  the  angle 
lohicli  this  p)erpendicidar  makes  with  the  axis  of  X. 

43.  Distances  from  a  Line  in  the  Normal  Form.  In  this 
form  A"^  -\-  B^  =  1.  Hence  the  distance  of  the  point  whose 
co-ordinates  are  x'  and  y'  from  the  line  is 

x'  cos  a  -\-  y'  sin  a  —  p 

the  same  function  which,  equated  to  zero,  represents  the  line. 
Hence  the  theorem: 

If,  in  the  expression  x  cos  a  -\-  y  sin  a  —  p,  we  suhstitiite 
for  X  and  y  the  co-ordinates  of  any  point  whatever,  the  expres- 


THE  STRAIGHT  LINE.  4,1 

sion  luill  represent  the  distance  of  that  point  from  the  line  whose 
equation  is  x  cos  oc  -\-  y  sin  a  —  p  =:  0. 

By  supposing  x'  and  y'  zero,  we  find  the  distance  of  the 
origin  from  the  line  to  be  —  p.  Since  p  itself  has  been  taken 
as  essentially  positive,  we  conclude: 

The  expression  for  the  distance  of  a  point  from  the  line  in 
the  normal  form  is  negative  luhen  the  point  is  07i  the  same  side 
as  the  origin,  and  positive  on  the  opposite  side. 

This  agrees  with  the  convention  that  the  direction /ro??i 
the  origin  to  the  line  shall  be  positive. 

EXERCISES. 

1.  What  is  the  relation  of  the  two  lines 

X  cos    30°  +  y  sin    30°  -  ^  =  0 
and  X  cos  210°  +  y  sin  310°  —  p  =  0? 

2.  Draw  approximately  by  the  eye  and  hand  the  lines 
represented  by  the  following  equations: 

X  cos  30°  +  y  sin  30°  -  5  =  0. 
X  cos  60°  +  y  sin  36°  -  5  =  0. 
X  cos  120°  +  y  sin  120°  -5  =  0. 
X  cos  210°  +  y  sin  240°  -5  =  0. 

Lines  Determined  by  Given  Conditions. 

When  a  line  is  required  to  fulfil  certain  conditions,  those 
conditions  must  be  expressed  algebraically  by  equations  of  con- 
dition involving  the  parameters  of  the  line.  The  values  of  the 
parameters  are  to  be  eliminated  from  the  equation  of  the  line 
by  means  of  these  equations  of  condition. 

Since  two  conditions  determine  a  line,  it  will  be  convenient 
to  employ  a  general  form  of  the  equation  of  the  line  in  which 
only  two  parameters  appear.     Such  a  form  is 

y  =  mx  +  h.  (a) 

44.  To  find  the  equation  of  a  line  ivhich  shall  pass  through 
a  given  p)oint  and  malce  a  given  angle  with  the  axis  of  X. 

Let  {x',  y')  =  the  given  point,  and 

f  =  the  given  angle. 


48 


PLANE  ANALYTIC  GEOMETRY. 


One  equation  of  condition  is  then 

m  =  tan  e, 

which  determines  the  parameter  ??^.  This  gives,  for  the  equa- 
tion of  the  line, 

y  =  X  tan  £  -{-  h.  (b) 

The  condition  that  the  line  shall  pass  through  the  point 
{x',  y')  is 

?/'  z=  mx'  -\-  h. 

To  eliminate  hy  we  subtract  this  equation  from  {b)  after 
substituting  the  value  of  7n.     This  gives 

y  —  y'  —  tan  ^  {x  —  a:'), 

which  is  the  required  equation  of  the  line  passing  through  the 
point  {x' ,  y')  and  making  an  angle  e  with  the  axis  of  X,  If 
we  write  m  for  tan  e,  it  becomes 


or 


y  -  y 

mx  —  y 


-  m(x  —  x'), 
mx'  +  ^/  =  0, 


(c) 


which,  compared  with  the  general  form 
Ax  -\-  By  -\-  C  =  0, 

^=  -  1; 


gives 


m: 


C  =  -  77ix'  4-  y\ 


45.   To  find  the  equation  of  a  line  passing  through  tiuo 
given  2Joints. 

Let  {x^y  y^  and  (x^,  yj  be  the  two  given  points. 
To   determine   the  param- 


eters  m  and  b,  we   have  the 

conditions 

y.  = 
y.= 

mx^  +  b,  I 
mx^  -\-b,) 

(d) 

which  give, 

by  subtraction 

> 

y.-y^ 

=  (^a  -  ^d'tn  ; 

•whence    m 

_y^-  y. 

X^   —  X, 

(e) 

THE  STRAIOHT  LINE.  49 

Subtracting  the  first  equation  of  {d)  from  (a),  we  have 

y  -y.  =  H^  -  ^,), 
and  substituting  the  vahie  of  m  gives 

2/  -  y.  =  ■F^f"(^  -  *.)'  (16) 

*^2     —     ^1 

which  is  the  required  equation  in  which  the  parameters  m  and 
h  are  replaced  by  the  co-ordinates  of  the  given  points. 

To  reduce  the  equation  to  the  general  form,  we  have,  by 
clearing  of  denominators, 

■A  =  y.-  .Vi;  ) 

B=x^-  X,',  \      (17) 

G  =  yX^2  -  ^i)  -  ^i(^3-  y.)  =  ^^y-  ^.y^- ' 

Remark.  Most  of  the  special  forms  of  the  equation  already  given 
are  cases  in  which  the  line  is  determined  by  given  conditions.  For 
example: 

In  the  form  (1)  (§  33)  the  given  quantities  are  the  slope  and  the  inter- 
cept on  the  axis  of  X 

In  the  form  (2)  they  are  the  slope  and  the  intercept  on  the  axis  of  Y, 

In  the  form  (3)  they  are  the  two  intercepts. 

In  the  Normal  form  they  are  the  length  of  the  perpendicular  from 
the  origin  upon  the  line,  and  the  inclination  of  the  perpendicular  to 
the  axis  of  X 

EXERCISES. 

1.  Write  the  equation  of  a  line  passing  through  the  point 
(—1,  2)  and  making  an  angle  of  135°  w^ith  the  axis  of  X. 

2.  Write  the  equation  of  a  line  passing  through  the  point 
(4,  —1)  and  making  an  angle  of  30°  with  the  axis  of  X;  find 
the  intercepts  which  it  cuts  off  from  the  axes,  and  the  ratios 
of  these  intercepts  to  the  length  of  the  line  included  between 
the  axes. 

3.  Find  the  equation  of  the  line  passing  through  the  points 
(2,  4)  and  (3,  —2),  and  find  its  intercepts  on  the  axes,  the  angle 
which  it  makes  with  the  axis  of  X,  and  its  distance  from  the 
origin. 

4.  Find  the  equation  of  the  line  making  an  angle  of  150° 
with  the  axis  of  X  and  passing  at  a  perpendicular  distance  5 
from  the  origin. 


50  PLANE  ANALYTIC  GEOMETRY. 

5.  Find  the  equiiuxon  of  die  line  passing  through  the  point 
(1,  5)  and  intercepting  a  length  3  on  the  axis  of  Y. 

G.  Find  the  equation  of  the  line  passing  through  the  point 
(5,-1)  and  intercepting  a  length  —  3  on  the  axis  of  Y. 

7.  Write  the  equations  of  lines  passing  through  the  three 
following  pairs  of  points: 

I.  The  points  {a,  h)  and  {a,  —  b), 
II.  The  points  (—  a,  h)  and  {a,  h). 
III.  The  points  («,  h)  and  (—  a,  —  h). 

8.  What  is  the  distance  from  the  point  (1,  5)  to  the  line 
joining  the  points  (—  3,  3)  and  (1,  6)?  Ans.  — . 

0 

9.  If  the  Yertices  of  a  triangle  are  at  the  points  (1,  3), 
(3,  —  5)  and  (—  1,  —  3),  write  the  equations  of  the  three 
sides  in  the  general  form,  and  find  the  distance  at  which  each 
side  passes  from  the  origin. 

10.  Write  the  equations  of  the  three  medial  lines  of  this 
last  triangle. 

Note.  A  medial  hue  of  a  triangle  is  the  line  from  either  vertex  to 
the  middle  of  the  opposite  side. 

11.  Given  the  co-ordinates  of  the  vertices  of  a  triangle, 
find  the  equations  of  the  lines  which  join  the  middle  points 
of  any  two  sides,  and  show  that  these  joining  lines  are  parallel 
to  the  sides  of  the  triangle. 

12.  Find  the  equations  of  the  three  sides  of  the  triangle 
whose  vertices  are  at  the  points  {a,  h),  (a',  I')  and  a",  h"). 
Then  find  the  product  of  the  length  of  each  side  into  its  dis- 
tance from  the  opposite  vertex,  and  show  that  each  of  these 
products  is  equal  to  the  double  area  of  the  triangle. 

First  write  the  general  equation  of  each  side,  using  the  form  (17). 
Then  note  the  relation  between  each  value  of  ^/A^  +  B'^  and  the  corre- 
sponding side  of  the  triangle.    Then  form  the  products  and  note  §  22. 

13.  Show  analytically  that  if  a  series  of  parallel  lines  are 
equidistant,  they  contain  between  them  equal  segments  of  the 
axes  of  co-ordinates.  Note  that  the  values  of  ^)  for  such  lines 
are  in  arithmetical  progression. 


THE  STRAIGHT  LINE.  61 


Relation  of  Two  Lines. 

46.  Problem.  To  express  the  angle  between  two  lines  in 
terms  of  the  parameters  of  the  lines. 

Let  the  lines  be 

Ax  -\-  By  -^  C  =  0  I  .  . 

and  A'x  +  B'y-\-  G'=  0.  i  ^""^ 

The  angle  between  them  will  be  the  difference  of  the  angles 
which  they  make  with  the  axis  of  X]  that  is,  using  the  pre- 
vious notation,  it  will  be  £  —  e\ 

The  expression  for  the  tangent  oi  s  —  e'  will  be  the 
simplest.     We  have,  by  trigonometry, 

.       .  ,,        tan  s  —  tan  f'  ,._. 

Substituting  the  values  of  tan  e  and  tan  a'  found  from 
(13),  this  equation  becomes,  by  reduction, 

tan  (.  -  O  ==  5^Tqr^-,.  (19) 

Or,  if  we  put,  as  before, 

m  E  tan  e,       m'  =  tan  b\ 
the  expression  will  be 

tan  (*  -  £')  =  '^^^^-r  (30) 

1  +  mm'  ^     ' 

Either  of  the  forms  (18),  (19)  and  (20)  is  a  solution  of  the 
problem. 

47.  The  following  are  special  cases  of  the  preceding 
general  problem: 

I.   To  find  the  condition  that  two  lines  shall  ie  parallel. 
This  condition  requires  that  we  have 

f  -  £'  =  0°    or    180°; 
that  is, 

tan  {e  —  a')  —  0. 


52  PLANE  ANALYTIC  GEOMETRY. 

Hence,  from  (20),  the  required  condition  is 
A'B  -  AB'  =  0, 

A^       A    [  (31) 


or  jy,  =  -jj. 

II.  To  find  the  condition  that  two  lines  shall  he  perpen- 
dicular to  each  other. 

The  lines  will  be  perpendicular  when 

£  -  e'  =  ±  90°. 
Then 

tan  (£  —  6')  =  00 . 

In  order  that  the  second  members  of  either  of  the  equa- 
tions (19)  or  (20)  may  become  infinite,  its  denominator  must 
be  zero.     Hence  we  must  have 

AA^  +  BB'  =  0,  )  .^^. 

or  1  +  mm'  =  0,  )  ^     ' 

or  tan  e  tan  e'  =  —  1, 

*  which  are  three  equivalent  forms. 

EXERCISES. 

Write  the  equations  of  the  lines  passing  through  the  origin 
and  perpendicular  to  each  of  the  following  lines: 

1,  ax  -{■  by  -{-  c  =  0.        Ans.  hx  —  ay  —  0. 

2.  y  =  mx  -\-  h.  3.  a{x  -\-  y)  —  h(x  —  y)  =  0. 
4.  X  +  ny  =  c.  5.   (x  -  x^)  =  7)i{y  -  y^). 

6.  Write  the  equation  of  the  line  passing  through  the  point 
(a,  h)  and  perpendicular  to  the  line 

Ax  +  %  +  C  =  0. 

7.  Write  the  equation  of  the  line  through  the  point  {a,  b) 
parallel  to  the  line 

Ax-{-  By  -{•  C  =  0. 

8.  Express  the  tangent,  sine  and  cosine  of  the  angle 
between  the  lines 

ax  -\-  by  -\-  c  =  0; 
ax  —  by  '\-  c  =  0, 


THE  STRAIGUT  LINE.  53 

9.  Write  the  equations  of  two  lines  passing  through  the 
origin,  and  each  making  an  angle  of  45°  with  the  line 

ax  -\-  hy  -\-  c  =^  0, 

Ans.      (a  -\-  b)  X  —  {a  —  I))  y  —  0, 
and    {a  —  b)  X  -{-  {a  -\-  b)  y  =  0. 

10.  Compute  the  interior  angles  of  the  triangle  the  equa- 
tions of  whose  sides  are 

a;  -  3y  +  7  =  0; 
X  ~\-  y  —  d  =  0; 
X  -{-dy  —  0. 

11.  If  the  co-ordinates  of  the  three  vertices  of  a  triangle 
are  (2,  5),  (2,  —  3),  (4,  —  1),  it  is  required  to  find  the  equa- 
tions of  the  three  perj^endiculars  from  the  vertices  upon  the 
opposite  sides. 

12.  Find  the  equations  of  the  perpendicular  bisectors  of 
the  sides  of  the  same  triangle. 

13.  Show  that  the  lines  joining  the  middle  points  of  the 
consecutive  sides  of  a  quadrilateral  form  a  parallelogram. 

To  do  this  assume  symbols  for  the  co-ordinates  of  the  four  vertices; 
then  express  the  middle  points  of  the  sides  by  §  23,  and  then  the  equa- 
tions of  the  joining  lines  by  §  45,  and  show  that  opposite  lines  are 
parallel. 

14.  Find  the  condition  that  the  lines 

X  cos  a  -\-  y  mi  a  —  p  =  0      and      x  sin  /3  ^  y  cos  />  —  ^;' 
may  be  parallel. 

15.  If  two  lines  intersect  each  other  at  right  angles,  and  if 
a  and  b  be  the  intercepts  of  the  one  line,  and  a'  and  b'  of  the 
other,  it  is  required  to  show: 

(a)  That  of  the  four  quantities,  a,  b,  «',  &',  either  three 
will  be  positive  and  one  negative,  or  three  negative  and  one 
positive. 

(/?)  That  these  quantities  satisfy  the  condition 
aa'  +  bb'  =  0. 

16.  "What  is  the  rectangular  equation  of  the  line  whose 
polar  equation  is 

n 

-  =  4  COS  0  +  3  sin  6? 


54  PLANE  ANALYTIC  GEOMETRY. 

17.  Find  the  area  of  the  triangle  formed  by  the  straight 

lines 

y  =  X  tan  75°,        y  =  x,        y  =  x  tan  30°  +  2. 

18.  Kcduce  3r  cos  6  —  2r  sin  ^  =  7  to  the  form 

r  cos  (6  —  a)  =  ]?, 

and  find  the  values  of  a  and  ^j. 

19.  Show  that  if  F  -  a'  =  1,  the  lines 

X  -\-  (a-\-b)y+c  =  0    and     {a  +  b)x  -f-  {a'  -  h')y  -\- d  =  0 

are  perpendicular  to  each  other. 

20.  Show  that  when  the  axes  are  oblique,  the  ratio  x  :  y 
of  the  two  co-ordinates  of  a  point  is  equal  to  the  ratio 

Dist.  from  axis  of  Y :  Dist.  from  axis  of  X. 

21.  Show  that  the  lines  x  -\-  y  =  a  and  x  —  y  —  a  are 
at  right  angles,  whatever  be  the  axes. 

22.  Show  that  the  locus  of  a  point  equidistant  from  two 
straight  lines  is  the  bisector  of  the  angle  they  form. 

48.  To  find  the  point  of  intersection  of  two  lines  given 
hy  their  equations. 

As  already  shown,  the  co-ordinates  of  the  point  of  inter- 
section are  those  values  of  x  and  y  which  satisfy  loth  equations, 
(§  32).     If  the  given  equations  are 

Ax  -f  %  -f-  (7  =  0, 
A'x-\-B'y-^C'  =  Q, 
"we  find,  for  the  values  of  the  co-ordinates, 
_  BC'  -  B'O 
^-  AB'  -  A'B' 
_A'C  -  AC 
^  ~  AB'  -  A'B' 
which  are  the  required  co-ordinates  of  the  point  of  intersec- 
tion. 

Remark.  The  preceding  result  affords  another  way  of 
deducing  the  condition  of  parallelism  by  the  condition  that  two 
lines  are  parallel  when  their  point  of  intersection  recedes  to 
infinity.     Let  the  student  find  this  as  an  exercise. 


THE  STBAIOHT  LINE.  55 

49,  To  find  the  condition  that  three  straight  lines  shall 
intersect  in  a  poi7it. 

The  required  condition  must  be  expressed  in  the  form  of 

an  equation  of  condition  between  the  nine  parameters  of  the 

three  lines.     Let  the  equations  of  the  lines  be 

ax    -{-by    -\-  c    =0; 

a'x  +  h'y  +  c'   =0; 

a''x  +  h''y  +  c"  =  0. 

If  the  three  lines  intersect  in  a  point,  there  must  be  one 
pair  of  values  of  x  and  y  which  satisfy  all  three  equations. 
By  the  last  section  we  have,  for  the  co-ordinate  y  of  the  point 
of  intersection  of  the  first  two  lines, 

_  a'c  —  ac' 
y  "~  aV  -  a'V 
and  of  the  last  two, 

_  a"c'  -  a'c" 
y  -  a'h"  -  a"V 
If  the  three  lines  intersect  in  a  point,  these  values  of  y 
must  be  equal.     Equating  them  and  reducing,  we  find 

c{a'l"  -  a"V)  +  c\a"l  -  ah")  +  c'\ah'  —  a'h)  =  0, 
which  is  the  required  equation  of  condition. 

EXERCISES. 

1.  Given  the  three  lines 

x-\-2y^4:  =  0, 

2x-    y  -H  =  0, 

dx-^   y-\-c  =  0, 

it  is  required  to  determine  the  constant  c  so  that  the  lines 

vshall  intersect  in  a  point,  and  to  find  the  point  of  intersection. 

Ans.  c  =  ~  3. 

Point  =  (2, -3). 

2.  Express  the  condition  that  the  three  lines 

y  =  mx    +  c, 
y  z=  m'x  +  c', 
y  =  m"x  -j-  c", 
shall  intersect  in  a  point. 


66  PLANE  ANALYTIC  GEOMETRY. 

3.  Find  the  point  of  intersection  of  the  two  lines 

y  =  mx  -\-  c, 
y  =  m'x  —  c. 

4.  Prove  algebraically  that  if  two  lines  are  each  parallel  to 
a  third,  they  arc  parallel  to  each  other. 

Note.    We  do  this  by  showing  that  from  the  equations 

ah'    -  a'b    =  0, 
ab"   -  a"b   =  0, 
follows 

a'b"  -  a"h'  =  0. 

5.  If  the  equations  of  the  four  sides  of  a  parallelogram 
are 

y  =  mx   +  Cy 
y  =  m'x  —  c, 
y  =  mx  -\-  c\ 
y  =  m'x  —  c', 
it  is  required  to  find  the  co-ordinates  of  its  four  vertices  and 
the  equations  of  its  diagonals. 

Ans.)  in  part.     Equations  of  diagonals: 

m  4-  m' 


m   —  m 
6.  What  relation  must  exist  among  a,  a',  m  and  m'  that 
the  lines 

y  =  mx   -\-  a, 
y  =  m'x  —  a', 

may  intersect  on  the  axis  of  X?  Ans.  a' in  +  am'  =  0. 

50.   Transformation  to  New  Axes  of  Co-ordinates. 

By  the  formuloB  of  §  25,  the  equation  of  a  line  referred  to 
one  system  of  co-ordinates  may  be  changed  to  another  system 
by  an  algebraic  substitution. 

To  make  the  change  it  is  necessary  to  express  the  co- 
ordinates of  the  original  system  in  terms  of  those  of  the  new 
system,  and  to  substitute  the  expressions  thus  found  in  the 
equation  of  the  locus. 


THE  STRAIGHT  LINE. 


57 


0 


u 


M 


Example  I.    Let 

Ax-\-By-\-  C=0  {a) 

be  the  equation  of  a  line  referred  to 
the  system  {X,Y), 

Let  it  be  required  to  refer  the  line 
to  a  system  (X',  F')  parallel  to  the 
first  and  having  the  origin  0'  at  the 
point  (a,  h). 

By  §  26,  the  expressions  for  the  original  co-ordinates  in 
terms  of  the  new  ones  will  be 

X  —  X*  -f-  a\ 
y  =  //'  +  b. 

Substituting  these  values  in  the  equation  (a),  we  find  for 
the  equation  of  the  line,  in  terms  of  the  new  co-ordinates, 
Ax'  +  By'  -^Aa  -{-  Bb  -\-  C  =  0. 

The  coefficients  A  and  B  of  the  co-ordinates  remain  un- 
changed, showing  that  the  line  makes  the  same  angle  with  the 
new  axes  as  with  the  old  ones. 

Example  II.  Let  the  new  system  of  co-ordinates  have 
the  same  origin,  but  a  different  direction. 

The  equations  of  transformation  are  then  (3)  of  §  27. 
Substituting  the  values  of  x  and  y  there  given  in  the  equation 
(a)  of  the  preceding  example,  we  have 

(A  cos  d  +  ^  sin  S)x'  -{- {B  cos  S  -  A  sin  S)y'  +  C  =  0. 

The  sum  of  the  squares  of  the  coefficients  of  a;'  and  y' 
reduces  to  A^  -\-  B^,  as  it  should. 


EXERCISES. 

1.  What  will  be  the  equation  of  the  line 

y  =  2x  -^  5 
when  referred   to   new  axes,  parallel  to  the  original   ones, 
having  their  origin  at  the  point  (2,  3)? 

2.  What  change  must  be  made  in  the  direction  of  the  axis 
of  JTthat  the  line  whose  equation  isx  =  y  may  be  represented 
by  the  equation  x'  =  2y'  ? 


68 


PLANE  ANALYTIC  GEOMETRY. 


Section  II.    Use  of  the  Abbreviated  Notation.* 

51,  Functions  of  the  Co-ordinates.  We  call  to  mind  that, 
corresponding  to  any  point  we  choose  to  take  in  the  plane, 
there  will  be  a  definite  value  of  each  of  the  co-ordinates  x 
and  y.  Hence  if  we  take  any  function  of  x  and  y,  such,  for 
example,  as 

P  =  ^  +  2^  +  1, 
this  function  P  will  haye  a  definite  value  for  each  point  of 
the  plane,  which  value  is  formed  by  substituting  in  P  the 
values  of  the  co-ordinates  for  that  point.  We  may  then  im- 
agine that  on  each  point  is  written  the  value  of  P  correspond- 
ing to  that  point. 

Example. 

3  4  5  6  7  8 


3  4  5  6 

12  3  4 


-3        -2 


-1 


0 


-5        -4 

The  above  scheme  shows  the  values  of  the  preceding  func- 
tion P  =  a;  -f  2?/  +  1  for  a  few  equidistant  points,  assuming 
the  common  distance  between  the  consecutive  numbers  on 
each  line  to  be  the  unit  of  length. 

53.  Isorro2)ic  Lines.  We  may  imagine  lines  drawn 
through  all  points  for  which  P  has  the  same  value,  and  may 
call  these  lines  isorroinc;  that  is,  lines  of  equal  value.  We 
now  have  the  theorem: 


*  This  section  can  be  omitted  without  the  student  being  thereby  pre- 
vented from  going  on  with  subsequent  chapters.  But,  owing  to  the 
elegance  of  the  abbreviated  notation,  the  subject,  which  is  not  at  all 
abstruse,  is  recommended  to  all  having  mathematical  taste. 


THE  STRAIGHT  LINE. 


59 


If  the  function  P  is  of  the  first  degree  in  x  and  y,  the 
isorroinc  lines  will  form  a  system  of  j^rallel  straight  lines. 
Proof.     Let 

P  ^ax  -\-  by  -{•  c; 

and  let  us  inquire  for  what  points  P  has  the  constant  value 
k.  These  points  will  be  those  whose  co-ordinates  satisfy  the 
condition 

P  -  k  =  0, 
or 

ax  -{-  by  -\-  c  —  k  =  0.  {a) 

This  equation,  being  of  the  first  degree,  is  the  equation  of 
a  straight  line,  whose  angle  with  the  axis  of  X  is  given  by 
the  equation 

tan  €  = 7. 

0 

Since  a  and  b  retain  the  same  values,  whatever  values  we 
assign  to  h,  e  has  the  same  value  for  each  line  of  the  system, 
and  all  the  lines  are  parallel. 

53.  Distance  betiveen  Two  Lilies  of  the  System.  To  each 
value  of  h  in  the  equation  {a)  will  correspond  a  certain  line. 
We  now  have  the  problem: 

To  find  the  distance  betiueen  the  tioo  lines  for  which  P  has 
the  resjjective  values  ^,  and  h^. 


ir\ 


Let  OJf  and  OJVbe  the  respective  intercepts  of  the  lines 
on  the  axis  of  X.     We  shall  then  have 


Distance  MQ  =  MN  ^m  s. 


60  PLANE  ANALYTIC  OEOMETRT. 

Patting  ?/  =  0  in  the  two  equations 

ay  -]-hx^c-k^  =  0  \ 
and  ay-\-hx-{-c  —  lc^  =  Qf,) 

we  have,  for  the  intercepts, 

OM      ^'^^ 


MN=  ON-  0M  =  ^^-, 
and  MQ  =  {K  -  K)  %^  =  -^A..         (§36) 

Hence,  the  distance  apart  of  two  isorropic  lines  is  propor- 
tional to  the  difference  between  the  vahies  of  P. 

54.  Distance  of  a  Point  from  a  Line.  Let  us  now  re- 
turn to  the  general  expression 

P  =  ax  -{-  by  -\-  c,  (a) 

and  let  us  study  its  relation  to  the  line 

ax  +  by  -\-  c  =  0;)  ... 

that  is,  to  the  line  P  =  0.  )  ^  ^ 

In  (a)  we  may  suppose  x  and  y  to  have  any  values  what- 
ever. But  in  (b)  X  and  y  are  restricted  to  those  values  which 
correspond  to  the  different  points  of  the  line  («,  b,  c). 

Now  from  what  has  just  been  shown  it  follows  that  the 
points  for  which,  in  («),  P  has  the  special  value  h  all  lie  on 
a  straight  line  parallel  to  the  line  P  =  0,  and  distant  from  it 
by  the  quantity 

k 


Va'  +  b' 

Hence,  if  .t„  and  y^  be  the  co-ordinates  of  any  point  at 
pleasure,  we  have 

Distance  of  point  (x^,  y^)  from  line  (a,  b,  c)  =  ^'  "  "*     •^°  "^  ^, 

Va"^  ->-  b* 
a  result  already  obtained  in  §  41. 


THE  STRAIGHT  LINE.  61 

These  results  may  be  summed  up  in  a  third  fundamental 
principle  of  Analytic  Geometry,  as  follows: 

If 

P  ^  ax  -\-  by  -\-  c 

be  any  function  of  the  co-ordinates  of  the  first  degree,  then — 

I.  To  every  point  on  the  plane  will  correspond  one  definite 
value  of  P. 

II.  T/iis  value  of  P  is  equal  to  the  perpendicular  distance 
of  the  point  from  the  line  P  =  0  multiplied  by  the  constant 
factor  Va'  +  l)\ 

If  the  expression  P  is  in  the  normal  form,  we  have 

«'  +  ^'  =  1,  (§  36) 

and  the  factor  last  mentioned  becomes  unity. 
Hence — 

III.  If  we  have  a  function  of  x  and  y  of  the  form 

X  Q,Q&  a  -\-  y  ^m  a  —  p^  Py 
this  function  will  express  the  perpendicular  distance  of  the 
point  tvhose  co-ordinates  are  x  and  y  from  the  line  P  ==  0. 

EXERCISES. 

1.  Let  the  student  draw  the  line 

2^:  -  Sy  +  1  =  0, 
and  let  him  compute  the  values  of  the  expression 

2a;  -  3?/  +  1 
for  a  number  of  points,  and  lay  them  down,  as  in  the  scheme 
of  §  51,  until  he  sees  clearly  the  truth  of  all  the  preceding 
conclusions. 

2.  Imagine  a  plane  covered  with  values  of  the  function 

P  ^  ax  -\-  by  -\-  Cy 
as  in  §  51.  Around  the  origin  as  a  centre  we  describe  a  circle 
of  arbitrary  radius,  and  on  its  circumference  mark  the  points 
where  the  values  of  P  which  it  meets  are  greatest  and  least. 
Show  that  all  points  thus  marked  lie  on  the  line  bx—  ay  —  0. 
Show  also  on  what  line  the  points  will  fall  if  the  centre  of 
the  circle  is  at  the  point  (^,  q). 


PLANE  ANALYTIC  GEOMETRY, 


Theorems  of  the  Intersection  of  Lines. 

55.  We  rej)resent  by  the  symbols  P,  P',  etc.,  Q,  Q\  etc., 
different  linear  functions  of  the  co-ordinates;  e.g., 

P    ^  ax    -{-by     -}-  c; 

P'  =  a'x   +  h'y   -f  c'; 

P"  =  a"x  +  b''y  +  c"; 

etc.  etc.  etc. 

Also,  we  shall  represent  by  the  symbols  i/,  if',  etc.,  N,  N', 
etc.,  such  functions  reduced  to  the  normal  form  in  which 

a"  -\-h'  =  1. 

Since  the  Jf's,  iV^'s,  etc.,  will  be  a  special  case  of  the 
P's,  §'s,  etc.,  every  theorem  true  of  all  the  latter  will  also  be 
true  of  the  former;  but  the  reverse  will  not  always  be  the 
case. 

The  line  corresponding  to  the  equation 

P  =  0 
may,  for  brevity,  be  called  the  line  P. 

56.  Theorem.    If 

P  =  0,        P'  =  0 

he  the  equaiions  of  any  tivo  straight  lines,  and  if  /i  and  v  he 
any  two  factors  lohich  do  not  contain  x  or  y,  then  the  equa- 
tion 

^P    J^ypf     =     0  (/,) 

will  he  that  of  a  third  straight  Ihie  passi7ig  through  the^Joint 
of  intersection  of  the  lines  P  and  P' . 

Proof  1.  By  substituting  in  {h)  for  P  and  P'  the  ex- 
pressions which  they  represent,  we  see  that  jj-P  -\-  vQ  is  a 
function  of  the  first  degree  in  x  and  y. 

Hence  (Z>)  is  the  equation  of  some  straight  line. 

2.  That  point  whose  co-ordinates  satisfy  both  of  the  equa- 
tions P  =  0  and  P'  =  0  must  also  give  yuP  -f  vP'  =  0, 
and  must  therefore  lie  on  the  line  {IS). 


THE  STBAIOUT  LHIK  63 

But  such  point  is  the  point  of  intersection  of  the  lines 
P  and  P'. 

Hence  the  point  of  intersection  lies  on  the  line  {h),  and 
{h)  passes  through  that  point.     Q.  E.  D. 

Corollary.  If  three  functions,  P,  P'  and  P",  are  so  re- 
lated that  toe  can  find  three  factors,  A,  yu  and  v,  lohich  satisfy 
the  identity 

\P  +  ywP'  +  vP"  =  0, 

then  the  three  lines  P  =  0,  P'  =  0  and  P"  =  0  intersect 
ill  a  point. 

For  we  derive  from  this  identity 

whence,  by  the  theorem,  P  passes  through  the  point  of  inter- 
section of  P'  and  P" . 

57.  Theorem.     Conversely, 

If  P  =  0,  P'  =  0  and  P"  =  0  are  the  eqiiations  of  three 
lines  intersecting  in  a  point,  it  aliuays  ivill  he  possible  to  find 
three  coefficients,  p.,  v  and  A,  such  that 

jxP  +  vP'  +  AP"  E  0. 

Proof.  1.  Let  the  values  of  the  three  functions  P,  P' 
and  P"  be 

P    ^ax     -{-  hy     -\-  c     =  0;  j 

P'  E  a'x   +  h'y   +  c'    =  0;  V  {a) 

P"  =  a*'x  +  V'y  +  c"  =  0.  ) 

2.  Let  us  now  suppose 

/i  E  ^"^>   -  aV',  K  {b) 

V  E  ah'     —  ft'Z*;     ) 
and  let  us  form  the  expression  AP  -(-  jiP'  -f-  i^P".     In  this 
expression  we  shall  have 
Coefficient  oi  x  =  a(a'V'  -  a"V)  +  a'{a"h  -  ah") 

^a'\ah'  -  a'h)  (e  0); 
Coefficient  of  ?/  =  h(ci'h"  -  a"h')  +  h\a"h  -  ah") 

-\-h"{ah'  -  a'h)  (eO); 
Absolute  term    =  c{a'h"  -  a"h')   -f  c\a"h  -  ah") 

+  c"{a'b'  -  a'h). 


64  PLANE  ANALYTIC  OEOMETRT. 

Because  the  three  lines  pass  through  a  point,  this  absolute 
term  is  zero  (§  49).  Hence  the  whole  expression  is  identically 
zero,  and  the  values  {b)  of  A,  ^i  and  v  satisfy  the  conditions 
of  the  theorem. 

EXERCISES. 

1.  Show  that  if  the  equations  P  =  0  and  §  =  0  are  so 
related  that  we  can  find  two  coefficients,  fx  and  r,  which  form 
the  identity 

^P-\-vQ  =  0, 

then  the  two  lines  P  and  Q  are  coincident.    (Comp.  §§52,  53.) 

2.  Having  the  two  lines 

y  —  mx  ^    a  —  0, 

y  +  mx  -j-  2a  =  0, 
it  is  required  to  find  the  equation  of  a  third   line  passing 
through  their  point  of  intersection  and  through  the  origin. 

Method  of  Solution.  Calling  the  given  expressions  equated  to  zero 
P  and  Q,  and  noting  that  the  equation  of  every  line  through  the  point 
of  intersection  may  be  expressed  in  the  form 

we  are  to  determine  the  quantities  jj-  and  v  so  that  this  line  shall  pass 

through  the  origin.     Hence  the  absolute  term  must  vanish.     This  gives 

the  condition 

/I  =  —  2r, 

the  value  of  v  being  arbitrary.     Substituting  this  value  of//,  and  divid- 
ing by  y,  "we  find  the  required  equation, 

y  —  Zmx  =  0. 

3.  Find  the  equation  of  a  line  passing  through  the  origin 
and  through  the  point  of  intersection  of  the  lines 

y  —  2x  —    a  =  0; 
y  ^2x  -\-  3a  =  0. 

4.  Find  the  equations  of  the  lines  making  angles  of  45° 
and  135°  respectively  with  the  axis  of  Xand  passing  through 
the  point  of  intersection  of  the  above  two  lines. 

58.  To  complete  and  apply  the  preceding  theory,  it  is 
necessary  to  distinguish  between  the  positive  and  negative 
sides  of  a  line.  If  distances  measured  on  one  side  are  positive, 
those  on  the  other  side  are  negative.     But  no  rule  is  possible 


THE  STRAIOET  LINE.  65 

for  the  positive  and  negative  sides  without  some  convention, 
because  the  function  P  may  change  its  sign  without  changing 
the  position  of  the  line.     For  example,  the  two  equations 

X  —  ny  -f-  7i  =  0, 
—  X  -\-  ny  —  h  —  (d, 

represent  the  same  line;  but  all  values  of  x  and  y  which  make 
the  one  function  equal  to  -f  P  will  make  the  other  equal  to 
—  P,  so  that  the  positive  and  negative  sides  of  the  lines  are 
interchanged  by  the  change  of  form. 

Now,  in  the  first  form,  the  distance  of  the  origin  from  the 
line  is 

h 


Hence, 

When  the  absolute  term  in  the  equation  is  ]JOsUive,  the x>osi- 

tive  side  of  the  line  is  that  on  ^ohich  the  origioi  is  situated,  and 

vice  versa. 

In  the  normal  form  the  absolute  term  is  negative.     Hence, 
In  the  normal  form  a  i)0sitive  value  of  the  function 

31  =  X  cos  oi  -\-  y  sin  a  —  ]) 

indicates  that  the  point  whose  co-ordinates  are  x  and  y  is  on 
the  opposite  side  of  the  line  from  the  origin ,  and  a  7iegative 
value  that  it  is  on  the  same  side  as  the  origin, 

59.  Theorem.    If 

if  =  0,        i\r  =  0 

are  the  equations  of  any  two  lines  in  the  nor7nal  form,  tlien 
the  equations 

if+JV=0,         Jf-iV=0, 

will  he  the  equations  of  the  bisectors  of  the  four  angles  ivhich 
the  lines  Hand  Nform  at  their  point  of  intersection. 

Proof.  1.  Because  the  functions  if  and  i\^are  in  the  nor- 
mal form,  they  represent  the  respective  distances  of  any  point 
from  the  lines  i/  =  0  and  iV  =  0.     (§  54.) 


66  PLANE  ANALYTIC  GEOMETRY. 

2.  Every  pair  of  co-ordinates  which  fulfil  the  condition 

M  ±  N=  0 
must  give 

M  =^  ±  N, 

so  that  the  point  which  they  represent  is  equally  distant  from 
the  lines  M  and  N. 

3.  By  geometry,  the  locus  of  the  point  equally  distant  from 
two  lines  is  the  bisectors  of  the  angles  formed  by  the  lines. 

Remark  1.  This  theorem  holds  equally  true  of  the  equations  of 
auy  two  lines  in  which  the  sums  of  the  squares  of  the  coefficients  of  x 
and  y  are  equal.    For  if,  in  the  equations 

P  ^ax  -{-by  -\-c  =0, 
P^a>x-\-  h'y  -f  c'  =  0, 

we  have  a'  +  ^*  =  ^"  +  ^'^  then,  by  §  53,  the  functions  Pand  P',  when 
not  restricted  to  zero,  express  the  distances  of  a  point  {x,  y)  from  the  re- 
spective lines  Pand  P',  multiplied  by  Va^  -j-  b'^  and  Va"^  ^  b"^  respec- 
tively. 

Now,  when  these  multipliers  are  equal,  every  point  whose  co-ordi- 
nates satisfy  the  equation 

(d  ±  a')  x  +  {b  ±  b')  y  -\-  c  ±  c'  =  0 
or 

P±  P  =0 

must  be  equally  distant  from  the  lines  Pand  P'. 

Remark  2.  The  equation 

M  -  N=0 

will  be  that  of  the  bisector  of  the  angle  in  which  the  origin  is  situated, 
and  of  its  opposite  angle;  while  the  equation 

M-{-N=0 

will  represent  the  bisector  of  the  two  adjacent  angles. 

EXERCISES. 

Find  the  bisectors  of  the  angles  formed  by  the  following 
pairs  of  lines: 

1.  a;  _  2?/  =  0  and        2x  —  y  =  0. 

2.  y  -\-  nx  —  c  =  0        and        717/  —  x  -\-  c  =  0. 

3.  Prove  the  theorem  of  geometry  that  the  two  bisectors 
of  the  angles  formed  by  a  pair  of  intersecting  lines  are  at 
right  angles  to  each  other. 


::::(  '«> 


THE  STRAIOUT  LINE.  67 

In  other  words,  if  the  functions  P  and  P'  are  such  that 
cCi  +  62  ==  a'i  _|_  j'2^ 

then  show  that  the  two  lines 

P  -f  P'  =  0        and        P  -  P'  =  0 
intersect  at  right  angles. 

4.  Show  that  if  iV  =  0  and  N'  =  0  are  the  equations  of 
two  lines  in  the  normal  form,  then 

XN  +  }iN'  =  0, 

XN  -  }iN' 

will  represent  the  loci  of  those  points  whose  distances  from  N 
and  N'  are  in  the  ratio  // :  A.  Also,  show  geometrically  that 
such  a  locus  is  a  straight  line. 

5.  In  the  preceding  exercise,  what  condition  must  the  co- 
efficients X  and  yu  satisfy  in  order  that  the  equations  {a)  may 
each  be  in  the  normal  form? 

60.  Applications  of  the  Preceding  Theorems.  The  pre- 
ceding theorems  enable  us  to  prove  with  great  elegance  the 
leading  theorems  of  the  intersections  of  certain  lines  in  a  tri- 
angle. 

I.  The  bisectors  of  the  interior  angles  of  a  triangle  meet  in 
a  2)oint. 

Proof.     Let 

X  =  0,         if  =  0,         N=0, 
be  the  equations  of  the  sides  of  the  triangle. 

We  suppose  the  origin  to  be  within  the  triangle,  because 
we  can  always  move  it  thither  by  a  transformation  of  co-ordi- 
nates. 

Then,  by  what  precedes, 

P    =  L  -  M=0, 

p'  =M-  ]sr=  0, 

P"^N-  L  =  0, 

will  be  the  equations  of  the  bisectors.  But  these  functions, 
P,  P'  and  P",  fulfil  the  identity 

p  _|_  P'  +  P"  =  0, 

and  reduce  to  the  form  §  56  when  we  suppose 
X  =  jii  =  V  =  1. 


PLANE  ANALYTIC  GEOMETRY. 


Hence  P,  P'  and  P"  all  pass  through  a  point. 

II.  The  hisedors  of  any  two  exterior  angles  and  of  the 
third  interior  angle  meet  in  a  point. 

Proof  The  equations  of  two  exterior  bisectors  and  of  the 
third  interior  bisector  are 


P  ^  L-\-  M 
P'  ^M-\-  N 
P"^  L-  N 


0; 
0; 
0; 


which  fulfil  the  identity 

P  -  P' 


P"  E  0. 


III.   Tlie  perpendiculars  from  the  three  vertices  of  a  tri- 
angle upon  the  opposite  sides  meet  in  a  point. 

Let  P  be  any  point  upon  the  perpendicular  from  Y  upon 
al3;  PM  X  Ya,  PN  ±  F/?;  and 
y  =  angle  aY/3.  ^ 

Then,  because  the  angles  PYa  ^^ 

and  a  are  complementary, 

PM=  PY cos  a,  /         PT 

PN  =  PYcos  p] 

PM :  PN  =  cos  <a:  :  cos  /?. 

Therefore,  if  the  equations  of  the  sides  Ya  and  Y/3  are 

N  =  0  :  N  =  0, 

then,  by  the  theorem  of  §  59,  Ex.  4,  the  equation  of  the  per- 
pendicular YP  will  be 

N  cos  p  ~  N  cos  a  =  0. 

In  the  same  way,  if  the  equation  of  ap  is  N"  =  0,  we 
shall  have,  for  the  equations  of  the  other  two  perpendiculars, 

N   cos  a  —  N'  cos  y  =  0; 
N'  cos  y  —  N    cos  /3  =  0. 

The  sum  of  these  three  equations  is  identically  zero,  thus 
showing  that  the  three  lines  intersect  in  a  point. 


THE  STRAIGHT  LINE. 


69 


61.  Diagonals  of  a  Quadrilateral.  An  elegant  and  in- 
structive application  of  the  preceding  theory  is  given  by  the 
following  problem: 

To  find  the  cquatioiis  of  the  diagonals  of  a  quadrilateral  of 
luhich  the  equations  of  the  four  sides  are  given. 

We  remark  that,  in  general  geometry,  a  quadrilateral  has  three 
diagonals.  The  reason  is  that  each  side  is  supposed  to  be  of  indefinite 
length,  and  so  to  intersect  the 
three  others.  A  diagonal  is 
then  defined  as  the  line  joining 
the  point  of  intersection  of  any 
two  sides  to  the  point  of  in- 
tersection of  the  other  two 
sides.  The  number  of  points 
of  intersection,  or  vertices,  is 
equal  to  the  combinations  of 
two  in  four,  or  6.  Taken  in 
pairs  these  6  points  have  three 
junction  lines,  as  shown  in  the 
figure. 

Solution.    Let  the  equations  of  the  four  sides  be 
P  =  0: 

Q  =^\ 

R  =  0: 
S  =0.. 

We  seek  for  four  factors,  k,  A,  yu  and  v,  by  which  to  form 
the  identity 

kP  -{-\Q-\-  ^xR^  v8^  0.  {h) 

Four  such  factors  can  always  be  found  when  the  parame- 
ters of  Py  Q,  etc.,  are  given,  because  by  equating  to  zero 
the  coefficients  of  x  and  g  and  also  the  absolute  term  in  (b) 
we  shall  have  three  equations  which  determine  any  three  of 
the  four  factors  jc,  A,  /x  and  v  in  terms  of  the  fourth.  To 
the  latter  we  may  assign  any  value  at  pleasure. 

The  identity  (b)  being  satisfied,  we  shall  have 

hP-{-XQ=-  (m^  +  vS).  (c) 

Now,  (§  56), 

H,P  -irXQ  =  0 


(a) 


70  PLANE  ANALYTIC  GEOMETRY. 

is  the  equation  of  some  line  passing  through  the  intersection 
of  P  and  Q,  while 

fxR  J^  vS  =0 
is  the  equation  of  some  line  passing  through  the  intersection 
of  R  and  S. 

But,  by  (c),  these  two  lines  are  identical.  Hence  this  com- 
mon line  is  a  diagonal  of  the  quadrilateral.  We  show  in  the 
same  way  that 

kP  ^  lxR  =  0        or        XQ-\-  vS  =0 
is  the  equation  of  the  diagonal  joining  the  intersection  of  P 
and  R  to  that  of  Q  and  8.     Also,  that 

kP  -{-  yS  =0        or        \Q-\-  ^R  =  0 
is  the  equation  of  the  diagonal  joining  the  intersection  of  P 
and  S  to  that  of  Q  and  R. 

Example.  To  find  the  equations  of  the  diagonals  of  the 
quadrilateral  whose  sides  are 

P=    X  -{-    2/4-1  =  0: 

Q~    a;  +  2?/  -  3  =  0: 

R=    .T  -  2?/  +  4  =  0; 

^  =  2.^'  -    ?/  -  2  =  0. 

Forming  the  expression  (Z>),  we  find  it  to  be 

(;i+;i+/^+2y)a;+(7i+2;i-2//-^)?/  +  «-3A+4/^-2rE0. 

Hence,  to  form  this  identity,  (§  8), 

(1)  ;^  +    A  +      /I  4-  2r  =  0; 


(3) 

«  + 

2A  - 

2jx  -     V 

=  0; 

(3) 

H   — 

3A  4- 

4//  -  2v 

=  0. 

We  solve 

as  follows: 

(3)- 

-(1) 

X  - 

3;(  -  3v 

=  0; 

(3)- 

-(3) 

5A  - 

3A  4- 

6/^4-    ^ 
7t^  =  0; 

=  0. 

9/1  4-  16r  =  0. 

A  = 

7 
-3"  = 

21 

fX  = 

16 

-  ¥  "' 

H   = 

19 

THE  STllAIOHT  LINE.  71 

The  value  of  v  is  arbitrary,  and  values  of  u,  pt  and  A,  free 
from  fractions,  are  obtained  by  putting  y  =  9.  The  values 
of  the  four  coefficients  are  then 

;^  =  19;         A  =  -  21;         yu  =  -  16;         t-  =  9. 

From  these  coefficients  the  equations  of  the  diagonals  are 
formed  by  the  preceding  formulae,  and  are  found  to  be 

2.r  +  %Zy  -  82  =  0; 

x-\-Yty  -\h  ^  0; 

37a:  +  \^y  +1  =  0. 

63.  Fundamental  Lines  of  a  Triangle.  Let  us  consider 
the  following  problem: 

If  the  equations  of  the  three  sides  of  a  triangle  in  the  nor- 
mal form  are 

M    =  0, 

M"  =  0, 

what  line  is  represented  hy  the  equation 

M -\- M' -\- M"  =  0?  {a) 

Solution,     If  we  put 

Q  =  M-\-  M', 

the  equation  §  =  0  will  represent  the  bisector  of  the  exterior 
angle  between  the  lines  if  and  M'  (§  59). 
Also,  the  equation 

Q^M"  =  0, 

which  is  the  same  as  {a),  will  represent  some  line  passing 
through  the  point  of  intersection  of  if"  and  Q,  that  is, 
through  the  point  in  which  the  bisector  meets  the  opposite 
side. 

In  the  same  way  it  may  be  shown  that  the  line  («)  passes 
through  each  of  the  other  two  points  in  which  the  bisectors 
of  the  exterior  angles  meet  the  opposite  sides. 

Hence  the  solution  of  the  problem  leads  to  the  theorem : 
TJie  three  points  in  which  the  bisectors  of  the  exterior 


72  PLANE  ANALYTIC  GEOMETRY. 

angles  of  a  triangle  meet  the  op2)Osite  sides  lie  in  a  straight  line, 
namely,  the  line  lohose  equation  is 

M-\-  M'  -^r  M"  =  0; 

31  =  0,  3r  =  0  and  M"  =  0  beiiig  the  equations  of  the  sides 
in  the  normal  form. 

We  may  show  in  the  same  way  that  the  three  equations 

M  -^  M'  -  M''  =  0, 

J/  -  if'  +  if"  =  0, 

-M  +  if  +  M"  =  0, 

are  the  equations  of  three  straight  lines  each  containing  the 
foot  of  one  bisector  of  an  exterior  angle  and  two  bisectors  of 
the  two  remaining  interior  angles. 

EXERCISES. 

1.  Show  by  the  preceding  theorems  that  if  we  form  a 
triangle  by  joining  the  points  in  which  each  bisector  of  an 
interior  angle  meets  the  oi:>posite  side,  the  sides  of  this  tri- 
angle will  severally  pass  through  the  points  in  which  the 
bisectors  of  the  exterior  angles  meet  the  opposite  sides. 

2.  Show  that  if 

31  =  0,        3r  =  0,        3r'  =  0,         M'"  =  0, 

be  the  equations  of  the  four  sides  of  a  quadrilateral  in  the 
normal  form,  then 

3/  _|_  3f'  _|_  J/"  -I-  if'"  =  0 

will  be  the  equation  of  a  straight  line  containing  the  three 
points  in  which  the  external  bisectors  of  the  three  pairs  of 
opposite  vertices  meet  each  other. 

3.  Find  the  equations  of  the  three  diagonals  of  the  quad- 
rilateral whose  sides  are 

y  =  x; 

y  =  X  -{-h'y 
X  =  a; 
y  =  -  X. 


CHAPTER     IV 

THE  CIRCLE. 


Section  I.    Elementary  Theory. 


Eqviation  of  a  Circle. 

63.  Problem.     To  find  the  equation  of  a  circle,'^ 
Let  the  co-ordinates  of  the   T 

centre  G  of  the  circle  be  a 
and  i,  and  let  P  be  any  point 
of  the  circle. 

Calling  X  and  y  the  co- 
ordinates of  Py  we  have,  for 
the  square  of  the  distance  be- 
tween G  and  P, 

CP"  =  {x-  a)'  +  (y-  by.  (§  17) 

The  condition  that  P  shall  lie  on  the  circle  requires  that 
this  distance  shall  be  equal  to  the  radius  of  the  circle.  Let 
us  put 

r  E  GP,  the  radius  of  the  circle. 

The  condition  then  becomes 

(X  -  ay  -\-iy-  by  =  r%  (1) 

which  is  the  required  equation  of  the  circle. 

64.  Theorem.  Every  equation  letween  recta7igular  co- 
ordinates of  the  form 

H^'  +  f)  -{-px  +  qy-i-h  =  0  (2) 

*  In  the  almost  universal  notation  of  the  higher  geometry  the  word 
' '  circle"  is  used  to  designate  the  closed  curve  which,  in  elementary 
geometry,  is  called  the  circumference  of  the  circle. 


74  PLANE  ANALYTIC  GEOMETRY. 

in  tohich  the  coefficients  of  x^  and  if  are  equal,  tvhile  there  is 
no  term  in  xy,  represents  a  circle. 

Proof.     Diyide  by  m,  and  put,  for  brevity, 


a  =  - 

p 

b^  - 

2m' 

and  the 

equati 

ion  will  be  transformed  into 

x'  -  2ax  ■ 

i-f- 

.2by-{--  = 

0, 

or 

(x- 

■  «)'  +  (2/ 

-bf 

-  a'  -h'  ^ 

h 

or 

{X- 

■ «)'  +  (y 

-by 

=  a'  -^b'  - 

h 

0, 


(3) 

The  first  member  represents  the  square  of  the  distance  be- 
tween the  fixed  point  («,  b)  and  the  varying  point  {x,  y).  The 
second  member  being  a  constant,  the  equation  shows  that 
the  square  of  the  distance  of  the  two  points  is  a  constant, 
whence  the  distance  itself  is  a  constant.  Hence  the  equation 
represents  a  circle  whose  centre  is  at  the  point  (a,  b)  and 

whose  radius  is  y^^    \   y^  _  z^ 

m' 

65.  Special  Forms  of  the  Equation  oj  a  Circle, 

We  may  suppose  a  circle  moved  so  that  its  centre  shall 

occupy  any  required  position  without  the  form  or  magnitude 

of  the  circle  being  changed. 

If  the  centre  be  at  the  origin,  we  have  a  =  0  and  ^  =  0, 

and  the  equation  of  the  circle  becomes 

x^  -Vy'  =  r\  (4) 

If  the  centre  is  on  the  axis  of  X,  we  have  Z>  =  0,  and  the 
equation  becomes 

f^{x-  ay  =  r\ 

which  is  the  equation  of  a  circle  whose  centre  is  on  the  axis 
of  X. 


THE  CIRCLE. 


75 


If  we  suppose  a  =  r  and  5  =  0,  the  circle  will  be  tangent 
to  the  axis  of  Y  at  the  origin,  and  the  y 
equation  will  become 

=  2ax  —  x^,  (5) 

which  we  may  define  as  the  equation 
of  a  circle  when  a  diameter  is  taken  as 
the  axis  of  X  and  the  origin  is  at  the 
end  of  this  diameter. 


EXERCISES. 


Find  the  radii  and  the  co-ordinates  of  the  centres  of  circles 
having  the  following  equations : 


1.  x"    +    2/'  -  lO.r  4-    2y  +  17  =  0. 

2.  2,x'    +  3?/'  +    Qx  -  12?/  -    9  =  0. 

3.  2^:^    +  2if  +    8a;  -  18?/  -   I  =  0. 


4.  mx^  +  my''  +  P^'^  +  ([V 


2m 


5.  Write  the  equation  of  the  circle  whose  centre  is  at  the 
point  (1,  —  2)  and  whose  radius  is  7. 

6.  Write  the  equation  of  the  circle  whose  centre  is  in  the 
position  {p,  q)  and  whose  radius  is  Vp^  +  q". 

7.  Write  the  equation  of  the  circle  whose  centre  is  at  the 
point  (0,  5)  and  which  is  tangent  to  the  axis  of  X. 

8.  Write  the  equation  of  a  circle  passing  through  the 
origin  and  having  its  centre  at  the  point  (3,  4). 

9.  Find  the  equation  of  a  circle  of  which  the  line  drawn 
from  the  origin  to  the  point  {p,  q)  shall  be  a  diameter. 

10.  Find  the  equation  of  a  circle  of  which  the  line  from 
the  point  (1,  3)  to  the  point  (7,  —  5)  shall  be  a  diameter. 

11.  Find  the  locus  of  the  centre  of  the  circle  passing 
through  the  points  {p,  q)  and  (p',  q'),  and  show  that  it  is  a 
straight  line  perpendicular  to  the  line  joining  these  points. 


76  PLANE  ANALYTIC  GEOMETRY, 

MetJiod  of  Solution.  Since  the  two  points  are  to  lie  on  the  circle,  their 
co-ordinates  must  satisfy  the  equation  of  the  circle;  that  is,  we  must 
have 

(P   -  af  +  iq  -6)2  =  r^ 

ip'  -  ay  +  iq'  -  hf  =  r\ 
The  radius  r  being  a  quantity  which  must  not  appear  in  the  equation,  we 
must  eliminate  it,  which  we  do  by  mere  subtraction.  We  thus  find  au 
equation  of  the  first  degree  between  a  and  b,  the  co-ordinates  of  the 
centre.  To  express  the  locus  in  the  usual  form  we  may  write  x  and  y 
for  a  and  h  in  this  equation,  which  will  then  be  the  required  equation  of 
the  locus  of  the  centre. 

12.  Find  the  locus  of  the  centre  of  the  circle  passing 
through  the  points  (1,  1)  and  (7,  9). 

13.  Find  the  locus  of  the  centre  of  the  circle  passing 
through  the  origin  and  the  point  (p,  q). 

14.  Find  the  locus  of  the  centre  of  the  circle  passing 
through  the  origin  and  the  point   {2p  cos  a,  2p  sin  a). 

66.  Intersections  of  Circles.  The  points  in  which  circles 
intersect  each  other,  or  in  Avhich  a  straight  line  intersects  a 
circle,  are  found  from  the  values  of  the  co-ordinates  which 
satisfy  both  equations. 

Let  the  two  circles  which  intersect  be  given  by  the  equa- 
tions 

x'  -{-f  -i-ax   -j-hij    -^p    =0;)  ,  . 

^'  +  2/'  +  «'^'  +  ^'.y  +  ^/  =  0.  f  ^  ^ 

By  subtracting  one  of  these  equations  from  the  other,  we 
have 

(«  -  a')x  +  (b-  h')y  +  ;j  -  y  =  0; 

whence  y  = ^  /~  ^—, —. 

By  substituting  this  value  of  y  in  either  of  the  equations 
(a),  we  shall  have  a  quadratic  equation  in  x. 

Since  such  an  equation  has  two  roots,  there  will  be  two 
points  of  intersection. 

But  the  roots  may  be  imaginary.  The  circles  will  then 
not  meet  at  all,  but  one  will  be  wholly  within  or  wholly  with- 
out the  other. 

If  the  roots  are  equal,  the  points  of  intersection  are  coinci- 
dent, and  the  circles  touch  each  other. 


THE  CIRCLE.  77 


EXERCISES. 

1.  Find  the  points  of  intersection  and  the  length  of  the 
common  chord  of  the  two  circles 


a:'  +  I/'  -  rta:  =  d\ 

Ans,     The  co-ordinates  are: 

r^  —  d"^ 
X  = for  both  points; 

a 
Common  chord  =  -  |  a'r'-  (r'  -  dy  \  * 

2.  Find  the  points  of  intersection  and  tlie  length  of  the 
common  chord  of  the  circles 

x"  -\-y''  -a'^  0; 
x''^f-\-by -7-^  =  0, 

3.  Determine  the  radius  7'  so  that  the  circles 

x"  -Yy""  -2x  =  3, 

shall  touch  each  other. 

MetJiod  of  Solution.  We  find,  as  in  the  preceding  exercises,  the 
values  of  the  co-ordinates  x  and  y  of  the  points  of  intersection.  In  order 
that  the  roots  may  be  equal,  the  quantity  under  the  radical  sign  in  the 
expression  for  y  must  vanish.     Equating  it  to  zero,  we  shall  have 

7^  _  107'-'  +  9  =  0, 

an  equation  of  which  the  roots  are  3  and  1. 

4.  Find  the  distance  apart  of  the  two  points  in  which  the 
line 

x  =  y-\-l 
intersects  the  circle 

x"  -{-y'  =  10. 


78  PLANE  ANALYTIC  GEOMETRY. 

67.  Polar  Equation  to  the 
Circle. 

Let  0  be  the  pole,  OX  the 
initial  or  base  line; 

p'  and  a  the  i)olar  co-ordi- 
nates of  the  centre  C; 

p  and  6  the  polar  co-ordi- 
nates of  any  point  P. 

We  then  have,  by  trigonom-  o' 

etry, 

PC  =  OF'  +  OC  -  20P.  OC  cos  FOG; 
that  is,         r'  =  p'  +  p"  —  2pp'  cos  {d  —  a), 
or  p^  -  2pp'  cos  {6-  a)-\-  p"  -  r^  =  0,  (6) 

the  polar  equation  required. 

It  may  also  be  obtained  from  the  equation  referred  to 
rectangular  axes  by  putting  x  =  pcosO,  y  =  p  sin  0,  a  =  p' 
cos  ex,  and  b  =  p'  sin  a.  If  the  initial  line  pass  through  the 
centre,  a  =  0  and  the  equation  becomes 

p'  -  %pp'  cos  ^  4-  p'=  -  r=  =  0.  (7) 

If  the  origin  lie  on  the  circumference,  p''  =  r'  and  the 
equation  becomes 

p  =  2p'  cos  ^  =  2r  cos  0,  (8) 

Note.  In  the  above  we  put  p  and  p'  for  the  radii  vectores  in  order 
to  avoid  confusing  them  -with  the  radius  of  the  circle,  which  we  call  r. 

Tangents  and  Normals. 

68.  Equation  of  Tangent  to  a  Circle.  The  requirement 
that  a  line  shall  be  tangent  to  a  circle  does  not  alone  determine 
the  line,  because  a  circle  may  have  any  number  of  tangents. 
We  may  therefore  anticipate  that  this  requirement  will  be  ex- 
pressed by  an  equation  of  condition  between  the  parameters 
of  the  line.     Let  us  then  consider  the  problem: 

To  find  the  equation  -which  the  parameters  of  a  liiie  nmst 
satisfy  in  order  that  the  line  may  he  tangeiit  to  a  given  circle. 
Let  the  circle  be  given  by  the  equation 
(X  -  ay  +{y-  by  =  r\ 
and  let  the  equation  of  the  line  be 

Ax  +  %  +  C  =  0. 


THE  CIRCLE.  79 

By  geometry,  the  situation  of  the  line  must  be  such  tliat 
the  perpendicular  from  the  centre  of  the  circle  upon  it  shall 
be  equal  to  the  radius  of  the  circle.  Conversely,  every  line 
for  which  this  perpendicular  is  equal  to  the  radius  of  the 
circle  is  a  tangent. 

Now,  the  length  of  the  perpendicular  from  the  point  {a,  b) 
upon  the  line  {A,  B,  0)  is 

aA-^hB-^  Q 
VA^~+~B~' 

The  requirement  that  this  perpendicular  shall  be  equal  to 
the  radius  r  of  the  circle  gives  the  equation 

aA  +  bB-}-  C=rVA'  +  B%  (1) 

which  is  the  required  equation  of  condition  between  the  para- 
meters A,  B  and  C. 

If  the  equation  of  the  line  is  in  the  normal  form 
xcos  a  -{-  y  sin  a  —  2:^  =  0, 
we  shall  have 


VA'  -\-B'  =  Vcos'a  +  sin^a:  =  ±  1, 
and  the  equation  (1)  will  assume  the  form 

acos  a  -{-  b  sin  a  —  2^  =  ±  r,  (2) 

69o  Equation  of  the  tangent  expressed  in  terms  of  the  tan- 
gent of  the  angle  tuhich  the  line  mahes  ivith  the  axis  of  X. 
Let  y  —  mx  -J-  5  be  the  equation  of  the  tangent,  and 

the  equation  of  the  circle.     Eliminating  y  between  these  two 
equations,  we  have 

(1  +  m^)x^  +  2mbx  +  {W  -  r')  =  0, 
which  must  have  equal  roots,  since  the  tangent  touches  the 
circle  in  only  one  point.     Now  the  condition  that  this  equa- 
tion may  have  equal  roots  is  (§  8) 

m'b'  =  (l-\-m')  {b'  -r'); 
whence  b  =  r  Vl  +  m% 

which  substituted  in  the  equation  of  the  tangent  gives 

y  =  mx  ^r  Vl  -\-  m^  (3) 


80  PLANE  ANALYTIC  GEOMETRY. 

Conversely,  every  line  whose  equation  is  of  this  form  is  a 
tangent  to  the  circle. 

70.  Tangent  determined  by  Tivo  Conditions.  In  the 
preceding  article  we  employed  only  the  one  condition,  that  the 
line  should  be  tangent  to  the  circle.  Hence  the  line  could 
be  completely  found  only  when  one  of  the  parameters  was 
given.  In  order  to  determine  completely  the  tangent  line, 
some  other  condition  besides  its  tangency  to  the  circle  must 
be  given.     Examples  of  such  conditions  are: 

That  the  tangent  line  shall  touch  the  circle  at  a  given 
point; 

That  it  shall  pass  through  a  given  point  not  on  the  circle  ; 
That  it  shall  also  be  tangent  to  a  second  circle. 

71.  Problem.  To  fi^id  the  equation  of  the  line  tangent 
to  a  circle  and  passing  through  a  given  point. 

Let  ic'  and  y'  be  the  co-ordinates  of  the  given  point,  and 
X  cos  OL  -^  y  sin  a  —  p  =^  0 
the  equation  of  the  tangent.     Since  the  tangent  passes  through 
the  point  {x',  y'),  we  must  have 

x'  cos  a  -\-  y'  sm  a  —  p  =  0,  (4) 

which  combined  with  (2)  will  determine  the  two  parameters, 
a  and  p,  of  the  line. 

We  may,  however,  first  eliminate  p  by  subtraction,  wdiich 
gives  the  equation 

{a  —  x')  cos  a  -{-  {h  —  y')  sin  a  =  r. 

The    solution    of    this   equation,  which   is   obtained   by 

methods  given  in  trigonometry,  will  give  the  value  of  a.      It 

may  also  be  obtained  algebraically  by  substituting  for  cos  a 

its   equivalent,    Vl  —  sTnP  a,   or  for   sin   a    its  equivalent, 

Vl  —  cos""  a,  or  for  cos  a  and  sin  a   their  equivalents   in 

terms  of  tan  a,  viz., 

1  .  tan  a 

cos  a  =  -=:         sm  a  — 


Vl  +  tan^a  VI  +  tan^ 

In  either  case  we  shall  have  a  quadratic  equation,  the  un- 
known quantity  in  which  will  be  either  sin  a,  cos  a  or  tan  a. 


THE  CIRCLE. 


81 


If  we  write,  for  brevity, 

VI  =  a  —  a:', 
n  =  b  -if, 

the  solution  of  these  equations  will  give 


sm  a  = 


cos  a 


tan  a 


nr  ±  m,  Vm"^  -\-  n"^  —  r^ 


=F 

m'  -\- 

if 

mr 

n  Vm"" 

-\-n^ 

-r' 

± 

m'  + 

n' 

mn 

r  Vm' 

+  n^ 

-r' 

(5) 


r  —  n 
The  value  of  j9  by  (4)  is 

p  z=.  x'  cos  a  -\-  y'  sin  a, 

in  which  cos  a  and  sin  a  must  be  replaced  by  their  values 
given  above.  Substituting  the  values  of  cos  a,  sin  a  and  ;:> 
in  the  equation 

X  cos  OL  -\-  y  sin  a:  —  ^j  =  0, 

the  result  will  be  the  required  equation  of  the  tangent  passing 
through  the  point  (x\  y').  The  double  sign  shows  that  there 
may  be  tioo  tangents  drawn  to  a  circle  from  a  point  without 
it. 

Case  when  the  given  point  is  on  the  circle.    In  this  case  we 
shall  have 

rrv"  +  n"  -  r""  =  0, 

and  the  values  (5)  of  sin  a  and  cos  a  become 


sm  a  =  -; 
r 


cos  a 


m 


and 


p  = 


mx*  4-  '^y* 


Substituting  these  values  in  the  equation  of  the  tangent, 
it  becomes 

mx  -\-  ny  —  mx'  —  ny*  =  0, 
or  771  (x  —  x')  -\-  n  (y  —  y')  =  0. 


82 


PLANE  ANALYTIC  GEOMETRY. 


If  we  substitute  for  m  and  n  their  values,  the  equation  is 

{a  -  x')  {X  -  X')  ^{b-  y')  (y  -  y')  =  0. 

If  we  take  the  centre  of  the  circle  as  the  origin,  we  have 

a  =  Q,  b  =  0; 

and  because  the  point  (x',  y')  is  on  the  circle, 

x''  +  y''  =  r\ 

Making  these  substitutions,  the  equation  of  the  tangent 
assumes  the  simple  form 

x'x  +  y'y  =  r\  (6) 

72.  Def.  The  subtangent  of  a  curve  is  the  projection 
on  the  axis  of  X  of  that  portion  of  the  tangent  intercepted 
between  the  point  of  tangency  and  its  intersection  with  the 
axis  of  X. 

Thus,  if  CT  is  the  axis  of 
X,  MT  is  the  subtangent  corre- 
sponding to  the  tangent  PT. 

Length  of  the  Subtangent.  To 
find  the  length  of  MT,  we  find 
the  intercept  GT  on  the  axis  of 
X  and  subtract  CM,  the  abscissa 
of  the  point  of  contact. 

The  equation  of  the  tangent  P^is  (6) 

x'x  -\-  y'y  =  r'; 
and  when  ?/  =  0,  we  have 

r 

X  —  ~ 

X* 

Hence  we  have 


CT. 


MT  = 


r  —  X 


X    =  - 


X  X 

73.  Def.  The  normal  to  any  curve  is  the  perpen- 
dicular to  the  tangent  at  the  point  of  contact. 

Equation  of  the  Normal  to  a  Circle.  The  equation  of  the 
line  perpendicular  to 

x'x  -|-  y*y  =  r"^ 


THE  CIRCLE.  83 

and  passing  through  the  point  of  contact  {x^,  y')  is,  by  §47, 

7/' 

y  -y'  ^  '^A^  -  ^'). 

or  y'x  —  x'y  =  0, 

the  equation  required. 

The  form  of  this  equation  shows  that  every  normal  of  a 
circle  passes  through  the  centre — a  property  which  is  easily 
established  by  elementary  plane  geometry. 

The  length  of  the  normal  is  that  portion  of  the  line  in- 
cluded between  the  point  of  contact  and  the  axis  on  which  the 
subtangent  is  measured.  In  the  case  of  the  circle,  the  nor- 
mal is  constant  and  equal  to  the  radius. 

Def.  The  subnormal  of  a  curve  is  the  projection  of 
the  normal  on  the  axis  of  X. 

Thus,  C3I  is  the  subnormal  corresponding  to  the  point  P, 
and  in  the  circle  is  equal  to  the  abscissa  of  the  point  of  contact. 


EXERCISES. 

1.  Show  that  the  condition  that  the  line 
X  cos  a  -\-  y  sin  a  —  p  =  0 
shall  be  tangent  to  the  circle 

(x-ay-j-(y-by  =  a'+b'^ 


is  a  cos  a  -{-  b  s'm  a  =p  ±  Va^  -j-  b'\ 

2.  What  is  the  condition  that  the  line 

y  =  mx  -\-  c  ^ 
shall  be  tangent  to  the  circle  c  5? -(>«*+ ^)-4/V^T7T>i?>P^ 

(X  -  ir  +  (2/  +  2y  =  16? 

3.  What  must  be  the  value  of  c  in  order  that  the  equation 

x  +  y  =  c 

may  be  tangent  to  the  same  circle? 

Ans.  c=  -1  ±4:V2. 


84  PLANE  ANALYTIC  GEOMETRY. 

4.  What  must  be  the  value  of  711  in  order  that  the  line 

y  =  mx  +  6 
may  be  tangent  to  the  circle 

x"  -\-y'  =  16? 

2"* 

5.  In  the  last  example  show  that  we  get  the  same  an- 
swer for  the  line 

y  =  mx  —  6, 

and  explain  the  equality  by  geometric  construction. 

6.  What  must  be  the  value  of  the  radius  d  in  order  that 
the  circle 

(X  +  3)'  +  (2/  -  ir  =  d^ 

may  have  as  a  tangent  the  line 

^  =  22: +  5? 

Ans.  d  =  — -zr. 

7.  By  elementary  geometry,  two  circles  are  tangent  to  each 
other  when  the  distance  of  their  centres  is  equal  to  the  sum 
or  difference  of  their  radii.  By  means  of  this  theorem  write 
out  the  condition  that  the  circles 

{x-aY  -{-{y-hy  =r'    ;(^-^7N-'^*'~-Vr 

and  {x-ay-^(y-by  =  r''      -    ^  ±t^ 

shall  be  tangent  to  each  other. 

8.  Show  that  the  length  of  the  common  chord  of  the  circles 
whose  equations  are 

(X  -  2Y  ^{y-3Y  =  9 
and  (x  -  3)''  +  (?/  -  2)^  =  9 

is  VU. 

9.  Find  the  condition  that  tlie  circles 

{x  -  hy  +  (y-  ky  =  a' 
and  {x  -  hy  +  (2/  -  hy  =  a' 

may  touch  each  other. 

Ans.  a  =  —-=.  (Jr.  —  h). 


THE  CIRCLE.  85 

10.  Show  that  the  polar  equation 

p'  —  {a  cos  6  -\-b  sin  0)  p  =  p^ 

is  that  of  a  circle,  and  express  its  radius  and  the  position  of 
its  centre. 

11.  What  curve  does 

p  z=  a  C0&  (6  —  a)  -\-  b  cos  {0  —  /3)  -\-  c  cos  (^  —  k)  +  •  •  • 
represent? 

12.  A  point  moves  so  that  the  sum  of  the  squares  of  its 
distances  from  the  four  sides  of  a  rectangle  is  constant.  Show 
that  the  locus  of  the  point  is  a  circle. 

13.  Given  the  base  of  a  triangle  {2b)  and  tlie  sum  of  tliQ 
squares  on  its  sides  (2m^),  find  tlie  locus  of  the  vertex  when 
the  middle  point  of  the  base  is  the  origin. 

Ans.  x"  -f  ?/^  =1 11  f  —  W. 

14.  Given  the  base  ifi)  and  the  vertical  angle  [B)  of  a  tri- 
angle, find  the  locus  of  the  vertex  when  the  origin  is  at  the 
end  of  the  base. 

Ans.  x^  -\-  if  —  bx  —  by  cot  B  —  ^. 

15.  Show  that  if,  in  the  equation 

x'^J^ifJ^Ax-\-By^C^^, 
we  have 

4(7  >  A^  ^B\ 

the  circle  will  be  imaginary.     It  is  enough  to  show  that  the 
radius  is  imaginary. 

16.  Show  that  a  circle  may  be  defined  as  the  locus  of  a 
point  the  square  of  whose  distance  from  a  fixed  point  is  pro- 
portional to  its  distance  from  a  fixed  line. 

17.  Show  that  a  circle  is  the  locus  of  a  point  the  sum  of 
the  squares  of  whose  distances  from  any  number  of  fixed 
points  is  a  constant. 

18.  If  --  =  77,  show  that  the  circles 

a'       b' 

x^  -\-  y"^  -\-  ax  -\-by   =0, 

x^  +  2/'  +  «'^  +  ^'y  =-  0, 

touch  at  the  origin. 


86  PLANE  ANALYTIC  QEOMETRT. 

19.  Find  the  locus  of  a  point  wliose  distances  from  two 
fixed  points  have  a  fixed  ratio  to  each  other. 
"-  20.  Express  analytically  the  locus  of  a  point  from  which  a 
tangent  drawn  to  a  circle  will  have  a  fixed  length  t. 
^^  21.  Find  the  locus  of  the  point  from  which  two  adjoining 
segments  of  the  same  straight  line  shall  be  seen  under  the 
same  varying  angle.  In  other  words,  if  A,  B  and  C  are 
three  points  in  the  same  straight  line,  find  the  locus  of  the 
point  X  which  will  satisfy  the  condition 

I  Angle  AXB  =  angle  BXC. 


>f\j 


^  22^  If  the  equation  of  the  circle  x^  -{-  y"^  = 
to  another  system  of  co-ordinates  having  the  same  ax^  but  a 
different  direction,  show  analytically  that  the  equation  will 
not  be  altered. 

23.  Show  analytically  that  if  a  circle  cuts  out  equal  chords 
from  the  tv/o  co-ordinate  axes,  the  co-ordinates  a  and  b  of  its 
centre  will  be  equal.   t^M^x^ur  «w-c    «.  ^  «/  r  »■-  a '-  "y  *  >  Jt^-cl-^ 

24.  Find  the  equation  of  the  circle  which  passes  through 
the  three  fixed  points  (x^y^),  {x^ij^),  {x^y^). 

25.  Having  given  the  circle  x^  +  ^^  +  ^^x  —  6y  —  2  =  0, 
find  the  equation  of  its  two  tangents,  each  of  which  is  parallel 
to  the  straight  line  y  =  2x  —  7.  y  ^  2 ^  -/-  ,j  ±u^y 

26.  The  circle  x"^  -{-  y^  =  r"^  has  tangents  touching  it  at  the 
respective  points  {x^  yj  and  (x^  y^.  Express  the  tangent  of 
the  angle  formed  by  these  tangents.  ^I/x' "/--?':_ 

27.  A  line  of  fixed  length  slides  along  the  axes  of  coordi- 
nates  in  such  a  way  that  one  end  constantly  remains  on  each 
axis.     What  is  the  locus  of  the  middle  point  of  the  line? 

28.  Given  a  point  {a,  h)  and  a  finite  straight  line  whose 
length  is  c,  find  the  locus  of  the  point  whose  distance  from 
{a,  h)  is  a  mean  proportional  between  c  and  its  distance  from 
the  line  x  cos  ol  -\-  y  sin  a  ^=  p. 

■  29.  Having  given  the  equation  of  the  circle  y"^  =  2rx  —  x^, 
let  chords  be  drawn  from  the  origin  to  all  points  of  the  circle, 
and  let  each  of  them  be  divided  in  the  constant  ratio  m  :  n. 
It  is  required  to  find  the  locus  of  the  points  of  division. 

30.  The  same  thing  being  supposed,  the  chords,  instead 


THE  CIRCLE.  87 

of  being  divided,  are  each  doubled.     Find  the  locus  of  the 
ends. 

31.  On  each  radius  of  a  circle  having  its  centre  at  the 
origin  a  distance  from  the  origin  is  measured  equal  to  the  or- 
dinate of  the  terminal  point  of  the  radius  on  the  circle.  Find 
the  locus  of  the  point  where  the  measures  end.      ^^^y^  -  ^  ^ 

32.  The  same  thing  being  supposed,  take  on  each  radius 
a  point  at  a  distance  from  the  centre  equal  to  the  abscissa  of 

the  end  of  the  radius,  and  find  the  locus  of  this  point,    a  V  ^  V  /jc^ 

33.  Find  the  locus  of  the  point  from  which  two  circles  p'^z," 
will  subtend  the  same  angle;  that  is,  from  which  the  angle  ^  y 
subtended  by  the  pair  of  tangents  to  one  circle  is  equal  to 
that  subtended  by  the  pair  of  tangents  to  the  other  circle. 

34.  Find  the  equation  of  that  circle  which  passes  througli 
the  origin  and  cuts  off  the  respective  intercepts  J9  and  q  from 

the  positive  parts  of  the  axes  of  JTand  Y.   (x  -|j2-  -/■  c^~  \)^-  r"^ 

35.  Find  the  locus  of  the  point  the  sum  of  the  squares  of 
whose  distances  from  the  sides  of  an  equilateral  triangle  is       I 
constant,  and  show  that  it  is  a  circle.     (To  simplify  the  prob-  /  w 
lem,  let  the  base  of  the  triangle  be  the  axis  of  X.)  i^^^^^y^^-aQ  v-^ia'i 

36.  Find  the  polar  equation  of  the  circle  when  the  origin 

is  on  the  circumference  and  the  initial  line  a  tangent.    P=  it"  i^9- 

37.  A  line  moves  so  that  the  sum  of  the  perpendiculars 
AP  and  BQ  from  two  fixed  points,  A  and  B,  shall  be  a  con- 
stant.    Find  the  locus  of  the  middle  point  of  the  segment     >>^ 

pQ.  :tvr=  c^-t^r  ^JL: 

38.  The  straight  line  whose  equation  is  3^  +  5rc  -|-  19  ==  0  ^   ^ 
cuts  the  circle   y"" -\- x"  =  113   in   two  points.     Wliat   is   the  r*-/^ 
length  of  the  chord  which  the  circle  cuts  off  from  the  line?    -  //j-2i 

39.  Find  the  equation  of  the  straight  line  which  cuts  the 
circle  x^  -\-  y^  =  169  in  two  points  whose  abscissa3  are  re- 
spectively —  12  and  -f  7. 

40.  Find  the  equation  of  a  line  passing  through  the  point 
{x\  y')  and  forming  in  the  circle  x"  -\-  y""  =  r''  o,  chord  whose 
length  is  J.     -^c^^^+^x^.^-/.  =o    >  =  /r-  eL^   ^'vv^^-a^'j^^-^^o 

41.  Through  the  point  (x',  ?/'),  inside  the  same  circle,  a 
chord  is  to  be  drawn  which  shall  be  bisected  by  the  point. 
Find  the  equation  of  the  chord. 


/^ 


88  PLANE  ANALYTIC  QEOMETBT. 

Systems  of  Circles. 

*74:.  Let  us  consider  the  expression 

{X  -  ay  -^r{y-  bf  -  d\ 
which,  for  brevity,  we  shall  represent  by  P,  putting 
P  ^x"  -{-y'  -'Hax-  2by  -^  a"  -\- V  -  d'^ 

To  every  point  on  the  plane  will  correspond  a  definite 
value  of  P,  found  by  substituting  the  co-ordinates  of  such 
point  in  this  value  of  P, 

We  may  form  any  number  of  expressions  of  this  form, 
such  as 

P'   ^x'  -{-y'  -  2a' X  -  Wy  +  a'""  +  V   -  d'^-, 

P"  ^x'  ^y'  ~  2a"x  -  V)"y  +  a"''  +  V"  -  d'"; 

etc.  etc.  etc. 

In  general,  the  co-ordinates  x  and  y  which  enter  into  P 
will  be  considered  as  entirely  unrestricted,  in  which  case  P 
will  be  simply  an  algebraic  function  of  x  and  y. 

But  we  may  also  inquire  about  those  special  values  of  x 
and  y  which  satisfy  the  equation  P  —  0.  AVe  know  from 
§§  63,  64  that  the  points  corresponding  to  these  special 
values  of  x  and  y  all  lie  on  a  circle  of  radius  d,  having  its 
centre  at  the  point  {a,  h).     We  now  have  the — 

75.  Theorem.  77^6  value  of  P  for  any  jyoint  of  the  plane 
is  equal  to  tlie  square  of  the  tangent  from  that  point  to  the 
circle  P  =  0. 

Proof  Let  P  be  the  point  {x,  y),         ^^-r- 
and  let  O  be  the  centre  of  the  circle 
P  =  0,  which  is,  by  hypothesis,  the 
point  (a,  h).     We  then  have 

Cr  =  {x-  ay  +  (^  -  hy-, 
and  because  PTC  is  a  right  angle, 
PT'  =  GP'  -  CT' 

=  (^  -  ay  +(2/  -  by  -  d\ 


:^p 


THE  CIRCLE.  89 

which  is  the  vahie  of  the  function  P,  thus  proving  the 
theorem. 

Remark.  If  the  point  {x,  y)  is  taken  within  the  circle,  P 
will  be  negative,  and  the  length  of  the  tangent,  being  the 
square  root  of  P,  will  be  imaginary. 

76.  Theorem.  If  P  =  o  and  P'  =  0  are  the  eq^iaiions 
of  two  circles,  any  equatmi  of  the  form 

^P  +  vP'  =  0  {a) 

will  represent  a  third  circle  passing  through  their  points  of 
intersection. 

Proof.  We  first  show  that  {a)  is  the  equation  of  some 
circle.  Substituting  for  P  and  P'  their  values,  we  have  for 
the  equation  of  the  curve 

{}^-\-y){x''  +  f)  -  2{Ma  +  va')x  -  2(//J  +  vy)y 

+  /^(a^  -\-b'  -d')-\-  y{a"  +  h''  -  d'')  =  0. 

Here  the  coefficients  of  x^  and  y^  are  equal  and  there  is  no 
term  in  xy.  Hence  (§  64)  the  curve  represented  by  the 
equation  {a)  is  some  circle. 

Secondly,  the  co-ordinates  of  all  points  in  which  the 
circles  P  and  P'  intersect  must  satisfy  both  of  the  equations 
P  =  0  and  P'  =  0.     Hence  they  also  satisfy  the  equation 

//P  +  rP'  =  0, 

and  therefore  the  points  of  intersection  lie  on  the  circle  of 
which  the  equation  is  {a). 

Hence  this  circle  passes  through  the  points  of  intersection 
of  the  circles  P  and  P'.     Q.  E.  D. 

Cor.  The  curve  represented  by  (a)  depends  only  on  the 
ratio  of  the  factors  jj.  and  v,  and  remains  unchanged  when 
both  are  multiplied  by  the  same  quantity. 

By  assigning  different  values  to  the  ratio  /<  :  v,  we  may 
determine  as  many  circles  as  we  please  passing  through  two 
points. 

A  collection  of  circles  passing  through  two  points  is  called 
a  family  of  circles. 


90  PLANE  ANALYTIC  OEOMETRT. 

77.  Problem.  To  find  the  locus  of  tlie  -point  from  loliich 
the  tangents  to  tiuo  circles  shall  have  a  given  ratio  to  each 
other. 

Solution.  Let  P  =  0  and  P'  =  0  be  the  equations  of  the 
two  circles,  and  let  the  tangents  from  the  moving  point  be  in 
the  ratio  m  :  m'. 

The  square  of  the  tangents  will  then  be  in  the  ratio 
m^  :  m'^.  But  these  squares  are  represented  by  the  respective 
values  of  P  and  P'  corresponding  to  the  point  from  which 
the  tangents  are  drawn.  Hence  between  these  values  of  P 
and  P'  we  have  the  proportion 

P:  P'  =  nf  :  m'\ 
which  gives 

m''P  -  m'P'  =  0.  (b) 

Because  the  co-ordinates  of  the  point  from  which  the 
tangents  are  drawn  must  satisfy  this  equation,  this  equation 
is  that  of  the  required  locus. 

Comparing  with  §  76,  we  see  that  the  equation  is  of  the 
form  (a).     Hence: 

Theorem.  T7te  locus  of  the  point  from  lohich  the  lengths 
of  the  tangents  drawn  to  tivo  circles  have  a  constant  ratio  to 
each  other  is  a  third  circle,  passing  through  the  common  points 
of  intersection  of  the  first  two  circles,  and  therefore  a  third 
circle  of  the  same  family. 

78.  Tlie  Radical  Axis.  If  the  ratio  m  :  m'  is  unity,  the 
equation  {h)  will  reduce  to 

P  -  P'  =  0, 

or,  substituting  for  P  and  P'  their  values, 

2(a'  -  a)x  +  2(&'  -  h)y  +  a"  -  a'  +  h''  -V-d'^^d''^  0, 

which,  being  of  the  first  degree,  is  the  equation  of  a  straight 
line.  From  the  results  of  §  76,  this  line  must  be  the  common 
chord  of  the  two  circles.     Hence: 

Theorem.  Ilie  lociis  of  the  point  from  which  the  tangents 
to  two  circles  are  equal  is  the  common  chord  of  the  two  circles. 

Tliis  locus  is  called  the  radical  axis  of  the  two  circles. 


THE  CIRCLE,  91 


Imaginary  Points  of  Intersection. 

79.  Tlie  tliC(:>rcni  of  §  77  holds  equally  true  wlieilicr 
the  circles  F  and  P'  intersect  or  not;  that  is,  it  leads  to  a 
third  circle  passing  through  the  points  of  intersection  of  two 
circles,  even  2uhe7i  these  two  circles  do  not  intersect.  If  in  this 
last  case  the  third  circle,  which  we  may  call  P",  really  inter- 
sected either  of  the  others,  say  P',  this  result  would  be  self- 
contradictory.  For  in  such  a  case  the  circle  P  could  not  pass 
through  the  intersection  of  P'  and  P",  and  so  the  result  of 
the  theorem  would  be  false. 

But  if  the  point  of  intersection  of  P  and  P'  is  nowlwre, 
there  will  be  nothing  contradictory  in  the  result,  if  only  P" 
intersects  each  of  them  nowhere. 

Again,  in  §  78  we  have  found  a  perfectly  general  equation 
of  the  radical  axis  founded  on  the  definition  that  the  radical 
axis  is  the  line  joining  the  points  of  intersection  of  two  circles, 
•which  equation  gives  the  real  radical  axis  even  when  the  circles 
do  not  intersect. 

If  we  take  any  special  case,  w^e  shall  find  that  the  algebraic 
processes  are  the  same  whether  the  circles  do  or  do  not  really 
intersect:  only,  in  the  latter  case,  the  co-ordinates  of  the 
points  of  intersection  will  be  imaginary.  To  illustrate  this  in 
the  simplest  way,  take  the  two  circles 

{x  -  3)^  +  (?/  -  3)^  =  9. 
To  determine  the  points  of  intersection  we  must  find 
values  of  x  and  y  which  satisfy  both  equations.     The  second 
equation  is,  by  reduction, 

x'-\-if  -Qx-  G?/  +  18  =  9.  (a) 

Substituting  the  value  of  x^  +  t/'  =  1  in  the  first  member, 
we  find 

10      5 


Hence 


,       5  ,         ,       10       ,   25 


92  PLANE  ANALYTIC  QEOMETRT. 

By  substituting  for  ?/'  its  value  1  —  x",  we  find 


Completing  square. 


2                1/                                    O 

^   -3  ^=-9- 

i-e, 

,       5          25       25-32 
"^        3^"^36~       36       ~ 

7 
36' 

The  square  being  negative  shows  that  the  roots  are  imagi- 
nary.    The  solution  gives,  for  the  points  of  intersection, 

5  ±  V^^ 


X  — 


y 


6 

5  qp  4/- 


The  co-ordinates  x  and  y  being  imaginary,  the  circles  do 
not  really  intersect.  But  these  imaginary  values  of  the  co- 
ordinates satisfy  the  equations  of  both  circles  and  also  the 
equation  [b)  of  the  radical  axis,  as  we  readily  find  by  the 
calculation : 


5^ 

-  7  ±  10  V^ 

•  7 

9  ±  5  V- 

■  7 

5^ 

38 

_  7  If:  10  |/Z: 

~7 

18 
9  T  5  V- 
18 
5 
~3 

? 
^ 

5 

36 

±7  +  5:f  7 
6 

10 
~  6" 

) 

2/^ 
x-^y 

In  taking  the  sum  of  the  first  two  equations,  the  imaginary 
terms  cancel  each  other  and  we  have  x^  -{-  y"^  =  1. 

Subtracting  6  times  the  third  equation  we  satisfy  {a),  and 
the  third  is  identical  with  (5),  which  is  the  equation  of  the 
radical  axis. 

We  adopt  the  following  forms  of  language  to  meet  this 
class  of  cases: 

I.  An  imaginary  point  is  a  fictitious  point  which  we 
sup2?ose  or  imagine  to  be  represented  by  imaginary  co-ordi- 
nates. 

II.  When  imaginary  co-ordinates  satisfy  the  equation  of  a 


THE  CIRCLE.  93 

curve,  we  may  talk  about  the  corresponding  imaginary  points 
as  belonging  to  that  curve. 

III.  A  curve  may  be  entirely  imaginary. 

Example.     The  equjition 

x"  -{-if  -2x-2y=\  -3 
is  that  of  a  circle.     But  we  may  write  it  in  the  form 

{X  -  ly  +  (2/  - 1)'  =  - 1. 

The  first  member  is  a  sum  of  two  squares,  and  therefore 
positive  for  all  real  values  of  x  and  y,  while  the  second  mem- 
ber is  negative.  Hence  there  are  no  real  points  whose  co- 
ordinates satisfy  the  equation. 

EXERCISES. 

1.  If  we  take,  on  the  axis  of  X,  two  imaginary  points    ^ 
whose  abscissas  are  a  -f-  hi  and  a  —  hi  respectively,  find  the 
abscissa  of  the  middle  point  between  them. 

2.  Using  the  method  of  §  45,  find  the  equation  of  the 
line  joining  the  imaginary  points  whose  co-ordinates  are — 

1st  point:  x'   —  ci,  y'  ~  a  -\-  2ci; 

2d    point:  x/'  =  h -\-  ci,        y"  =  a  -{-2h  +  2ci; 

and  show  that  it  is  the  real  line 

y  =  2x  -^  a. 

3.  Find  the  equation  of  the  circle  whose  centre  is  at  the 
point  («,  2^)  and  which  cuts  the  axis  of  Xat  the  points  de- 
scribed in  Ex.  1. 

A71S.  (x  -  ay  +  (y  -  2hy  =  U\ 

4.  Find  the  equation  of  a  circle  belonging  to  the  family 
fixed  by  the  pair 

(x-%y  +  (y-5Y=    9, 
and  passing  through  the  origin. 
Ans,  •/<  =  20;  v=-  9;  Eq. :  ll(a;^-f  ?/')  -  124a:  -  30y  =  0. 


94 


PLANE  ANALYTIC  QEOMETBT. 


Section  II.     Synthetic  Geometry  of  the  Circle. 

Poles  and  Polars. 

80.  Let  there  be  two  points,  P  and  P',   on  the  same 
straight  line  from  the  centre  0  of  a  fixed  circle,  and  so  situ- 


ated that  the  radius  OR  shall  be  a  mean  proportional  between 
OP  and  OP'. 

Through  either  of  the  points,  as  P',  draw  a  line  Q  perpen- 
dicular to  the  radius.     Then 

The  line  Q  is  called  the  polar  of  the  point  P  with 
respect  to  the  circle,  and  the  point  P  is  called  the  pole  of  the 
line  Q  with  respect  to  the  circle. 

Had  we  drawn  the  line  through  P,  it  would  have  been  the 
polar  of  the  point  P',  and  P'  would  have  been  the  pole  of  the 
line  through  P. 

81.  The  following  propositions  respecting  poles  and 
polars  flow  from  these  definitions: 

I.  To  every  point  in  the  plane  of  the  circle  corresponds 
one  definite  polar,  and  to  every  line  one  definite  pole. 

II.  The  polar  of  a  point  and  the  pole  of  a  line  may  be 
found  by  construction  as  follows: 

io)  If  the  pole  P  is  given,  we  draw  the  radius  through 
the  pole  intersecting  the  circle  at  R.  We  then  find  the  point 
P'  by  the  proportion  OP  :  OE  =  OR  :  OP'. 

The  perpendicular  through  P'  will  be  the  polar  of  P. 

(b)  If  the  polar  is  given,  wc  draw  the  perpendicular  from 


TlIE  CIRCLE. 


95 


the  centre  0  upon  the  polar,  and  produce  it  if  necessary.  If 
it  intersects  the  circle  at  R  and  the  polar  at  P',  we  determine 
OP  as  the  third  proportional  to  OP'  and  OR.  The  point  P 
will  then  be  the  required  pole. 

III.  When  the  pole  is  within  the  circle,  the  polar  is  wholly 
without  it. 

IV.  If  a  pole  is  without  the  circle,  the  polar  cuts  the  circle. 

V.  When  the  pole  is  a  point  on  the  circle,  the  polar  is  the 
tangent  at  that  point. 

■    VI.  If  the  pole  ajiproaches  indefinitely  near  the  centre  of 
the  circle,  the  polar  recedes  indefinitely,  and  vice  versa. 

82.  Fundamental  Theorem.    If  a  line  jkiss  through  a 
point,  the  polar  of  the  point  ivillpass  through  the  pole  of  the  line. 

Proof.  Let  the  line  CD  pass 
through  the  point  P. 

By  definition,  we  find  the  polar 
of  P  by  drawing  the  radius  OM 
through  P,  taking  the  point  P'  so 
that,  putting  r  for  the  radius  OM, 

OP  '.r  =  r:  OP',     {a) 
aod  drawing         P' Q'  \_  0P\ 

We  find  the  pole  of  CD  by  draw- 
ing OQ  1_  CD  and  finding  a  point  P' 


such  that 


OQ:r  =  r'.  OP".  (b) 

We  have  to  prove  that  P"  lies  on  the  polar  P'Q'.  If  we 
call  Q'  the  point  in  which  OQ  meets  the  polar  P'Q',  the  tri- 
angles P'OQ'  and  QOP,  being  both  right-angled  and  having 
the  angle  at  0  common,  are  equiangular  and  therefore  similar. 

Hence 

OQ:  0P=  OP'  :  OQ'.  (r) 

Comparing  the  proportions  {a)  and  (h),  we  have 

OP.OP'  =  OQ.OP", 

which  gives  the  proportion 

OQ:  0P=  OP'  :  OP". 

Comparing  this  proportion  with  (c),  Ave  have 

OQ'  =  OP". 


96 


PLANE  ANALYTIC  OEOMETRT. 


Hence  P"  and  Q'  coincide;  that  is,  the  pole  P"  lies  on  the 
pohirP'(2'.     Q.  E.  D. 

Cor,  1.  We  may  imagine  several  lines  all  passing,  like  CD, 
through  the  point  F.  The  theorem  shows  that  the  poles  of 
these  lines  all  lie  on  P'Q\     Hence, 

If  several  lilies  j^^ass  through  a  point,  their  poles  luill  all  lie 
upton  the  polar  of  the  point. 


Cor.  2.  We  may  imagine  several  points,  all  lying,  like  P, 
on  the  line  CD.  The  theorem  shows  that  the  polars  of  these 
points  will  all  pass  through  Q',  the  pole  of  CD.     Hence, 

If  several  points  lie  in  a  straight  line,  their  polars  will  all 
pass  through  the  pole  of  the  line. 

Kemark.  These  several  theorems  may  be  more  readily 
grasped  when  placed  in  the  following  form: 

1.  If  a  line  turn  round  on  a  point,  its  pole  will  move  along 
the  polar  of  that  point. 

%.  If  a  p>oi7it  move  along  a  line,  its  polar  will  turn  round 
on  the  ptole  of  that  line. 

83.  Theorem  I.  If  from  any 
poi7it  tivo  tangeiits  he  draiun  to  a 
circle,  the  line  joining  the  points 
of  contact  will  he  the  polar  of  the 
pdhit. 

Proof.  Let  the  tangents  from 
P  touch  the  circle  at  M  and  iV. 
Let  Q  be  the  point  in  which  OP, 
from  the  centre  0,  intersects  the  line  i/iV. 


THE  CIRCLE.  97 

By  elementary  geometry,  OQN  and  ONP  are  right  tri- 
angles. Because  they  have  the  angle  at  0  common^  they  are 
equiangular  and  similar.     Hence 

OQ:  0N=  ON:  OP, 

Now,  since  OiVis  the  radius  of  the  circle,  this  proportion 
shows  that  MNh  the  polar  of  P.     Q.  E.  D. 

Theorem  II.  If  through  any  point  a  chord  he  drawn  to 
a  circle,  the  tangents  at  the  extremities  of  the  chord  will  meet 
071  the  polar  of  the  point. 

Proof.  Let  the  chord  pass 
through  the  point  Q,  and  let 
the  tangents  meet  at  P'.  By 
Theorem  I. ,  P'  is  the  pole  of  the 
chord;  therefore,  because  Q  lies 
on  the  chord,  the  polar  of  Q 
passes  through  P',  the  pole  of 
the  chord.     Q.  E.  D. 

Cor,  1.  If  any  number  of  chords  le  drawn  through  the 
same  point,  the  locus  of  the  point  in  which  the  tangents  at 
their  extremities  intersect  will  be  a  straight  line,  the  polar  of 
the  point. 

Cor.  2.  Conversely,  If  from  a  moving  point  on  a  straight 
line  tangents  be  draivn  to  a  fixed  circle,  the  chords  joining  the 
corresponding  points  of  tangency  will  all  pass  through  the  pole 
of  the  line. 

THEOREMS   FOR  EXERCISE. 

1.  If  we  take  any  four  points,  A,  B,  A'  and  B',  on  a  circle, 
and  if  P  be  the  point  of  meeting  of  the  tangents  at  A  and  B, 
and  P'  the  point  of  meeting  of  the  tangents  at  A'  and  B% 
then  the  point  of  meeting  of  the  lines  AB  and  A^B'  will  be 
the  pole  of  PP'. 

2.  If  we  take  four  points.  A,  B,  Xand  T,  on  a  circle,  such 
that  the  tangents  at  A  and  B  and  the  secant  XT  pass  through 
a  point,  then  the  tangents  at  Xand  l^and  the  secant  AB 
will  also  pass  through  a  point. 


98  PLANE  ANALYTIC  GEOMETRY. 


Centres  of  Similitude. 

84.  Def.  The  line  joining  the  centres  of  two  circles  is 
called  their  central  line. 


Theorem.  If  the  ends  of  parallel  radii  of  two  circles  he 
joined  ly  straight  lines,  these  lines  tuill  all  2^(^ss  through  a 
common  point  on  the  central  line. 

Proof.  Let  GP  and  C'P'  be  any  two  parallel  radii,  and 
8  the  point  in  which  the  line  PP'  intersects  the  line  CC 
joining  the  centres.  The  similar  triangles  SPG,  SP'G'  give 
the  proportion 

8G :  SG'  =  GP  :  G'P'.  (a) 

Putting,  for  brevity,  r  =  the  radius  GP,  and  r'  ~  the  radius 
C'P',  this  proportion  gives,  by  division, 

SG  -  SG'  :  SG'  =  GP  -  G'P'  :  G'P'  =  r-  r'  :r\ 
or  GG'  :  SG'  =  r-  r'  :  r'. 

Ilcnce  SG'  =-^,GG'; 

r  —  r 

that  is,  the  distance  SG'  is  equal  to  the  line  GG'  multiplied 
by  a  factor  which  is  independent  of  the  direction  of  the  radii 
GPf  G'P;  therefore  the  point  S  is  the  same  for  all  pairs  of 
parallel  radii.     Q.  E.  D. 

Gase  of  oppositely  directed  radii.      If  the  radii  CP,  G'P' 
be  drawn  in  opposite  directions,  it  may  be  shown  in  a  similar 


THE  CIRCLE.  99 

way  that  the  line  PP'  intersects  the  central  line  CC  in  a 
point  8'  determined  by  the  proportion 

CS'  :  C'S'  =  CP  :  C'P'  =  r  :  r',  {b) 

whence  S'  is  a  fixed  point  in  this  case  also. 


Def.  The  two  points  through  which  pass  all  lines  joining 
the  ends  of  parallel  radii  of  two  circles  are  called  the  cen- 
tres of  similitude  of  the  two  circles. 

The  direct  centre  of  similitude  is  that  determined 
by  similarly  directed  radii. 

The  inverse  centre  of  similitude  is  that  determined 
by  oppositely  directed  radii. 

Corollaries.  The  following  corollaries  should,  so  far  as 
necessary,  be  demonstrated  by  the  student. 

I.  The  direct  centre  of  sitliilitude  is  ahvays  ivitliout  the 
central  line  of  the  two  circles,  and  the  inverse  centre  is  always 
loithin  this  line,  hotuever  the  two  circles  may  he  situated. 

II.  If  the  tioo  circles  are  entirely  external  to  each  other,  the 
centres  of  similitude  are  the  points  of  meeti^ig  of  the  pairs  of 
common  tangents  to  the  two  circles. 

III.  Comparing  the  proportions  {a)  and  {h),  we  see  that  the 
point  S  divides  the  line  CC^  externally  into  segments  having 
the  ratio  r  :  r',  while  aS"  divides  it  internally  into  segments 
having  this  same  ratio. 

This  is  the  definition  of  a  harmonic  division.     Hence 
The  two  centres  of  similitude  divide  harmonically  the  li?ie 
joining  the  centres  of  the  tiuo  circles. 

85.  The  following  are  fundamental  theorems  relating  to 
centres  of  similitude: 

Theorem  I.  Every  line  similarly  dividing  tiuo  2)arallel 
radii  of  two  circles  passes  through  their  centre  of  similitude. 


100  PLANE  ANALYTIC  GEOMETRY. 

Proof.     Let  A  and  A'  be  the  points  in  which  the  line  di- 
vides the  radii; 

S'f  the  point  in  which  this  line  cuts  the  central  line; 
S,  the  centre  of  similitude; 
r,  r',  the  radii  of  the  circles. 


We  then  have,  by  the  property  of  the  centre  of  similitude, 

a  8  :  CO'  =  C'P'  :  CP  -  C'P'  =  r'  :r-r\        {a) 
The  similar  triangles  CAS'  and  C'A'S'  give 
C'8'  :  CS'  =  C'A'  :  CA; 
whence,  by  division, 

C'S'  :  CC  =  C'A'  :  CA  -  C'A\  (h) 

By  hypothesis,  the  radii  are  similarly  divided  at  A  and  A'; 

CA'  :  CA  =  r'  :r; 
whence,  by  division, 

CA'  :  CA  -  CA'  =  r\r-r'. 
Comparing  with  {a)  and  (b), 

CS'  :  CC  =  CS :  CC; 
whence  CS'  =  CS  and  the  points  S  and  S'  coincide.    Q.E.  D. 

Kemark  1.  If  the  radii  are  oppositely  directed,  the 
centre  of  similitude  will  be  the  inverse  one.  The  demonstra- 
tion is  the  same  in  principle. 

Eemark  2.  The  demonstration  may  be  shortened  by 
employing  the  theorem  of   geometry  that  there  is  only  one 


THE  CIRCLE.  -101 

point,  internal  or  externiil,  in  which  a  line  can  be  divided  in 
a  given  ratio.  (See  Elementary  Geometry.)  By  the  funda- 
mental property  of  the  centre  of  similitude,  it  divides  the 
central  line  into  segments  proportional  to  the  radii  of  the 
circles.  It  may  be  shown  that  the  point  ;S"  divides  the  central 
line  into  segments  proportional  to  CA  and  C'A'.  From  this 
the  student  may  frame  the  demonstration  as  an  exercise. 

Theorem  II.  Conversely,  If  any  line  pass  through  a 
centre  of  similitude,  and  j^ctt^^Uds  he  drawn  from  the  centres 
of  the  circles  to  this  line,  the  lengths  of  these  parallels  loill  he 
proportional  to  the  radii  of  the  circles. 


The  demonstration  is  so  easy  that  it  may  be  supplied  by 
the  student. 

86.  The  Four  Axes  of  Similitude  of  Three  Circles.  If 
there  be  three  circles,  they  form  three  pairs,  each  with  its 
direct  and  inverse  centre  of  similitude.  There  will  there- 
fore be  six  such  centres  in  all,  three  direct  ones  and  three  in- 
verse ones.     The  following  propositions  relate  to  this  case: 

Theorem  III.  The  three  direct  centres  of  similitude  lie 
in  a  straight  line. 

Proof.     Let  r^,  r,  and  r^  be  the  radii  of  the  three  circles. 

Let  8^,  S^  and  8^  be  the  direct  centres  of  similitude  of  the 
pairs  of  circles  (2,  3),  (3,  1),  (1,  2)  respectively. 

Let  a  line  AB  hQ  passed  through  8^  and  8^. 

From  the  centres  of  the  three  circles  draw  three  parallel 
lines,  i?j,  R^  and  B.^  to  the  line  AB.     Then, 

Because  AB  is  a  line  passing  through  the  centre  of  simili- 
tude 8^, 


lora 


PLANE  ANALYTIC  GEOMETRY. 


B,  :B,  =  r,:  r,.  (Th.  11.) 

Because  AB  isn  line  passing  through  /S,, 

B,  :  B,  =  r,  :  r,. 
Taking  the  quotients  of  these  ratios,  we  have 

B,:B,=  i\  :  r,; 
hence  the  line  AB  divides  the  radii  r,  and  1\  similarly. 


Therefore  this  line  passes  through  the  centre  of  similitude 
8,  of  the  circles  (1,  2).     (Th.  I.)     Q.  E.  D. 

Theorem  IV.  Each  direct  centre  of  similitude  lies  in  t^ie 
same  line  with  the  tiuo  inverse  centres  of  similitude  lohich  are 
7iot paired  luith  it. 

The  demonstration  of  this  theorem  is  so  nearly  like  that 
of  the  last  one  that  it  may  be  supplied  by  the  student. 

Def.  A  straight  line  which  contains  three  centres  of  simili- 
tude of  a  system  of  three  circles  is  called  an  axis  of  simili- 
tude of  the  system. 

Corollary.  For  each  system  of  tliree  circles  there  are  four 
axes  of  similitude,  of  which  one  contains  the  three  direct  cen- 
tres of  similitude,  and  the  others  each  contain  one  direct  and 
two  inverse  centres. 

EXERCISES. 

1.  Show  that  if  two  of  the  three  circles  be  equal,  two  of 
the  axes  of  similitude  will  be  parallel,  and  vice  versa, 

2.  If  all  three  circles  are  equal,  describe  the  axes  of  simili- 
tude. 


THE  CIRCLE. 


103 


The  Radical  Axis. 

87.  Theorem.  If  any  'perpendicular  ho  drawn  to  the 
ceiitral  line  of  tivo  circles,  the  difference  of  the  squares  of  the 
tangents  from  any  one  point  of  this  perpendicular  ivill  be  the 
same  as  from  every  other  point  of  it. 

Proof.  Let  PiV^  be  any  per- 
pend iculiir  to  the  central  line 
of  the  circles  C  and  6",  and 
P  any  point  on  this  perpen- 
dicular. 

Let  R  and  R'  represent  the 
distances  of  P  from  C  and  C\ 

Because  the  tangents  PT 
and  PT'  meet  the  radii  drawn 
to  the  points  T  and  T  of  con- 
tact at  right  angles,  we  have 

PT 

-prpn 


R'    -r'; 
R''  -  r'\ 


Hence,  for  the  difference  of  the  squares  of  the  tangents, 

PT^  _  PT'^  Z3  i^^  -  R'^  -  (r^  -  r'').  (1) 

From  the  right  triangles  PNG  and  PNC,  we  find,  in  the 
same  way, 

R^  -  R''  =  NC  -  NC"; 

whence,  from  (1), 

PT^  _  prpn  ^  jy-^2  _  j^fjn  _  ^^.2  _  ^«)  ^^j 

The  second  member  of  this  equation  has  the  same  value  at 
whatever  point  on  the  perpendicular  P  may  be  situated, 
which  proves  the  theorem. 

Corollary.  If  we  choose  the  point  i\^so  as  to  fulfil  the 
condition 


NC 


we  shall  have 


NO"  =  r' 


p/TT2  J) 'Jin 


(3). 


104  PLANE  ANALYTIC  GEOMETRY 

and  the  tangents  will  be  equal  from  every  point  of  the  per- 
pendicular, which  will  then,  by  definition,  be  the  radical  axis. 

SS,  Case  luhen  the  circles  intersect.  In  this  case  the 
tangents  drawn  from  either  point  of  intersection  are  both 
zero  and  therefore  equal.  Hence  this  point  is  on  the  radical 
axis,  and  this  axis  is  then  the  common  chord  (or  secant)  of 
the  two  circles.     Hence  another  definition : 

The  radical  axis  of  two  circles  is  their  common  chord, 
produced  indefinitely  in  both  directions. 

EXERCISE. 

In  the  preceding  construction  the  circles  have  been  drawn 
comi^letely  outside  of  each  other.  Let  the  student  extend 
the  general  proof  (1)  to  the  case  when  the  circles  intersect, 
showing  that  the  two  tangents  from  every  point  of  the  com- 
mon secant  are  equal,  and  (2)  to  the  case  when  one  circle  is 
wholly  within  the  other,  showing  that  the  radical  axis  is  then 
wholly  without  the  outer  circle. 

89.  The  Radical  Centre  of 
Three  Circles.  If  we  have  three 
circles,  each  of  the  three  pairs 
will  have  its  radical  axis.  We 
now  have  the  theorem: 

The  three  radical  axes  of  three 
circles  intersect  in  a  x^oint. 

Proof  Let^,  B  and  C  be  the 
three  circles,  and  let   0  be  tlie 

point  in  which  the  radical  axis  of  A  and  B  intersects 
radical  axis  of  B  and  G. 

Because  0  is  on  the  radical  axis  of  A  and  B, 

Tangent  0  to  ^  =  tangent  0  to  B. 
Because  0  is  on  the  radical  axis  of  B  and  C, 

Tangent  0  io  B  =  tangent  0  to  C. 

Hence     Tangent  0  to  ^  =  tangent  0  to  (7; 

whence  0  lies  on  the  radical  axis  of  A  and  C,  and  all  three 
axes  pass  through  0. 


THE  CIRCLE. 


105 


Def.  Tlie  point  in  which  the  three  radical  axes  intersect  is 
called  the  radical  centre  of  the  tliree  circles. 

Cor,  Tlie  radical  centre  of  three  circles  is  a  certai7i  point 
from  which  the  tangents  to  the  three  circles  are  all  equal. 

90.  System  of  Circles  having  a  Common  Radical  Axis* 
The  theory  of  a  family  of  circles,  developed  analytically  in  the 
preceding  section,  will  now  be  explained  synthetically. 

Peoblem.  Let  us  have  a  circle  A  and  a  straight  line  N: 
it  is  required  to  find  a  second  circle  X,  such  that  N  shall  be 
the  radical  axis  of  the  circles  A  and  X. 


Solution.  From  the  centre  A  draw  an  indefinite  line  AX 
perpendicular  to  the  line  N. 

Take  any  point  P  on  the  radical  axis  N,  and  from  it  draw 
a  tangent  PT  to  the  given  circle. 

From  the  same  point,  P,  draw  another  line,  PT,  in  any 
direction  whatever;  make  PT'  =  PT,  and  from  T'  draw  T'X 
perpendicular  to  PT'  and  meeting  the  central  line  in  X. 
The  circle  round  the  centre  X  with  the  radius  XT'  will  be 
that  required. 

For  PT',  being  perpendicular  to  the  radius,  is  tangent 
to  the  circle  X;  and  because  PT  =  PT',  the  line  through 
P  perpendicular  to  the  central  line  is  the  radical  axis. 
Hence  the  given  line  iVis  the  radical  axis  of  the  two  circles; 
whence  the  circle  JT  fulfils  the  condition  of  the  problem. 

Since  the  line  PT'  may  be  drawn  in  any  direction  what- 
ever, we  may  find  an  indefinite  number  of  circles  which  fulfil 
the  conditions  of  the  problem. 


10(5  PLANE  ANALYTIC  OEOMETRT. 

The  construction  of  these  circles  is  shown  in  the  figure. 
Since  the  tangents  from  P  are  all  equal,  it  follows  that  the 


line  PN  is  the  radical  axis  of  any  two  circles  of  a  family 
passing  through  the  same  two  points,  real  or  imaginary. 

Tangent  Circles. 

91.  The  following  propositions  lead  to  the  solution  of 
the  noted  problem  of  drawing  a  circle  tangent  to  three  given 
circles. 

Def.  When  two  circles  each  touch  a  third,  the  line 
through  the  points  of  tangency  is  called  the  chord  of  con- 
tact. 

When  two  circles  touch  each  other,  either  one  must  be 
wholly  within  the  other,  ot  each  must  be  wholly  without  the 
other.  Hence  contacts  are  said  to  be  of  two  kinds,  Mo'^nal 
and  external. 

Theorem  I.  First,  If  a  circle  is  tmlgent  to  a  pair  of 
other  circles,  the  chord  of  co7itact  passes  through  a  centre  of 
similitude  of  the  j)air. 

Secondly,  This  centre  of  similitude  is  the  direct  one  when 


THE  CIRCLE.  107 

the  contacts  are  of  the  same  hind,  and  the  inverse  one  ivhen 
they  are  of  opposite  kinds. 

Proof.  The  points  of  con- 
tact are  readily  shown  to  be  cen- 
tres of  similitude  of  the  respective 
pairs  of  tangent  circles. 

By  §  86,  any  two  centres  of 
similitude  of  different  pairs  lie 
on  a  straight  line  with  one  of  the 
centres  of  similitude  of  the  third  pair. 

Hence  the  points  of  tangency  are  in  the  same  line  with  a 
centre  of  similitude.     Q.  E.  D. 

Remark  1.  An  independent  proof  of  the  theorem  is  ob- 
tained by  drawing  the  radii  from  each  centre  of  the  pair  of 
circles  to  the  points  in  which  the  joining  line  intersects  the 
circumferences,  and  showing  that  the  radii,  taken  two  and 
two,  are  parallel. 

Remaek  2.  The  second  part  of  the  theorem  is  left  as  an 
exercise  for  the  student. 

02.  Homologous  Points.  If  a  common  secant  to  two 
circles  be  drawn  through  either  of  their  centres  of  similitude, 
it  will  intersect  each  circle  in  two  points.      By  combining 


either  of  these  points  on  one  circle  with  either  of  the  points 
on  the  other  circle  we  may  form  four  pairs  of  points,  as 
(P,  P').  {Q,  Q')^  (ft  n,  and  (P,  Q').  The  pairs  at  the 
termini  of  parallel  radii,  namely,  (P,  P')  and  (Q,  Q'),  are 
called  homologous  points;  those  at  the  termini  of  non- 
parallel  radii,  as  ((>,  P')  and  (P,  Q'),  are  called  anti-ho- 
mologous. 


108 


PLANE  ANALYTIC  GEOMETRY. 


93.  Theorem  II.  If  tivo  secants  he  drmvn  through  a 
centre  of  similittide,  then — 

I.  Tlie  distances  of  any  two  homologous  points  on  one  secant 
from  the  centre  of  similitude  ivill  he  proportional  to  the  dis- 
tances of  the  corresponding  points  on  the  other  secant. 

II.  The  products  of  the  distances  of  two  anti-homologous 
2mnts  loill  he  the  same  on  the  tiuo  secants. 


Hypothesis.     Two  secants,  SQ'  and  ST',  from  the  centre 
of  similitude  S,  cut  the  circles  in  the  points  P,  Q,  P'  and 
C,  and  R,  T,  R'  and  T  respectively. 
Conclusions : 

I.  SP  :  SP'  =  SR  :  SR'; 
SQ:  SQ'  =  ST:  ST. 
11.  SP .  SQ'  =  SR  .  Sr  =  SQ  .  SP'  =  ST.  SR'. 

Proof.  I.  Draw  the  central  line  and  the  radii  to  tlie 
points  of  intersection.  Because  of  the  parallelism  of  the  radii 
OP  and  OP',  etc.,  we  have 

Triangle  SOP  similar  to  triangle  SO'P' 
Triangle  SOQ  similar  to  triangle  SO'Q'; 
Triangle  SOR  similar  to  triangle  SO'R'] 
Triangle  SOT  similar  to  triangle  SO'T'. 

From  the  similarity  of  these  triangles,  we  have 

SO  :  SO'  --=  SP  :  SP'  =  SQ  :  SQ' 

=  SR  :  SR'  =  ST :  ST'.  Q.  E.  D. 

II.  The  second  and  last  of  these  proportions  give 


SP  .  SQ'  =  SP'  .  SQ; 
SR  .  ST'  =  SR'  .  ST. 


i\ 


(") 


THE  CIRCLE.  109 

By  a  fundamental  property  of  the  circle,  shown  in  elemen- 
tary geometry, 

SP  .  SQ   =  SR    .  ST; 

SF' .  SQ'  =  SR' .  sr. 

Multiplying  these  equations,  we  have 

SP  .  SQ'  X  SP'  .  SQ  =  SR.  sr  X  SR'  .  ST. 
By  substitution  from  (a),  this  equation  becomes 

(SP  .  ^'S')'  =  (^'^ .  ^^')'; 

whence,  extracting  the  square  root  and  combining  with  (a), 
we  have  conclusion  II.     Q.  E.  D. 

94.  Pef,  When  a  circle  touches  two  others,  we  call  it  a 
direct  tangency  when  the  two  tangencies  are  of  the  same  kind, 
and  an  inverse  tangency  when  they  are  of  opposite  kinds. 

Several  pairs  of  tangencies,  all  direct  or  all  inverse,  may 
be  called  of  the  same  nature.  If  one  pair  is  direct  and  another 
inverse,  they  are  of  opposite  natures. 

Eemakk.  It  will  be  noted  that  the  chords  of  contact 
pass  through  the  same  centre  of  similitude  in  the  case  of  two 
pairs  of  tangencies  of  the  same  nature,  but  not  otherwise. 
Hence,  in  what  follows,  whenever  we  have  several  circles 
touching  two  others,  we  shall  suppose  the  tangencies  to  be  of 
the  same  nature. 

95.  Theorem  III.  //'  each  circle  of  one  pair  is  a  tan- 
gent  of  the  same  nature  to  the  two  circles  of  another  pair,  then 
the  radical  axis  of  each  pair  passes  tltrough  a  centre  of  simili- 
tude  of  the  other  pair. 

Proof.  Let  the  circles  P  and  P'  touch  the  circles  0  and 
0'  at  the  points  M,  N,  M'  and  N'. 

The  point  of  meeting,  S,  of  the  lines  iVif  and  N'M'  will 
be  a  centre  of  similitude  of  0  and  0'  (§91).     Hence  we  have 

SM.  SN  =  SM' .  SN'.  (§  93) 

But  SM .  >S'iVis  equal  to  the  square  of  the  tangent  from  S 
to  the  circle  P  (EL  Geom.),  and  SM'.  SN'  is  the  square  of  the 


110  PLANE  ANALYTIC  GEOMETRY. 

tangent  from  8  to  tlie  circle  P'.     The  tangents  being  equal,. 
S  is  on  the  radical  axis  of  P  and  P\     Q.  E.  D. 


It  is  shown  in  the  same  way  that  a  centre  of  similitude  of 
P  and  P'  is  on  the  radical  axis  of  0  and  0\     Q.  E.  D. 

Cor.  1.  If  each  of  three  circles  is  a  tangent  of  the  same 
nature  to  two  other  circles,  then,  by  this  theorem,  one  of  the 
centres  of  similitude  of  each  two  out  of  the  three  circles  must 
lie  on  the  radical  axis  of  the  two  circles  which  they  touch. 
Hence, 

When  each  of  two  circles  touches  each  of  three  other  circles^ 
their  radical  axis  tvillform  one  of  the  axes  of  similitude  of  the 
three  circles. 

Cor.  2.  The  same  thing  being  supposed,  each  radical 
axis  of  the  three  circles  will,  by  the  theorem,  pass  through  a 
centre  of  similitude  of  the  pair  which  they  touch.  This  centre 
of  similitude  will  therefore  be  their  point  of  intersection. 
Hence, 

W7ie7i  each  of  two  circles  touches  each  of  three  other  circles, 
the  radical  centre  of  the  three  circles  will  I?e  a  centre  of  simili- 
tude of  the  two  circles. 

96.  Problem.  To  draw  a  circle  tangent  to  three  given 
circles. 

Construction.     Let  L,  TIf  and  Wbe  the  three  circles. 
Find  their  radical  centre,  (7,  and  an  axis  of  similitude,  S, 


TEE  CIRCLE. 


Ill 


Find  the  poles  p,  g,  r,  of  8  with  respect  to  the  three 
circles. 

Join  Cp,  Cq  and  6V,  and  let  ?,  ??i,  71  and  Z',  7?i',  n'  be  the 
points  in  which  these  lines  intersect  the  three  circles. 


The  circle  through  the  three  points  l,  m,  n  will  be  one  of 
the  tangent  circles  required,  and  the  circle  through  the  three 
points  Vy  m',  n'  will  be  the  other. 

Proof.  The  axis  of  similitude  8  is  the  radical  axis  of 
some  pair  of  circles  touching  the  three  given  circles  (§  95, 
Cor.  1),  and  C  is  one  of  their  centres  of  similitude  (§  95, 
Cor.  2). 

Let  us  call  X  and  Y  the  two  circles  of  this  pair,  which  are 
not  represented  in  the  figure.* 

Let  m,  I,  n  and  m',  n' ,  V,  instead  of  being  defined  by  the 
above  construction,  be  defined  as  the  points  of  tangency  of 
this  pair  of  circles  whose  centre  of  similitude  is  at  C.  Then, 
by  (§  91),  the  lines  mm',  nn'  and  IV  will  all  pass  through  C 

Through  m'  draw  the  common  tangent  to  the  circles  M 
and  X,  and  through  m  draw  the  common  tangent  to  J/ and 
Y,  and  let  P  be  the  point  of  meeting  of  these  tangents. 


*  The  tangent  circles  and  tangents  are  omitted  from  the  printed 
figure  to  avoid  confusing  it.  The  student  can  supply  them  so  far  as 
necessary. 


112  PLANE  ANALYTIC  GEOMETRY. 

Then,  because  the  tangents  Pm  and  Pm/  touch  the  same 
circle  M  at  m  and  m',  they  are  equal. 

Hence  these  lines  are  also  equal  tangents  to  the  circles 
X  and  Y]  hence  P  lies  on  the  radical  axis  of  Xand  Y,  that 
is,  on  the  line  S  (§11). 

Because  the  line  mm'  is  the  chord  of  contact  of  tangents 
from  Py  it  is  the  polar  of  P;  hence  the  pole  of  S,  a  line  through 
X,  lies  on  the  polar  mm'.  That  is,  the  line  Cq,  found  by  the 
construction,  passes  through  the  points  of  contact  m  and  m'. 

In  the  same  way  it  is  shown  that  the  points  of  contact  I,  V 
and  n,  n'  are  upon  the  lines  joining  C  and  the  poles  j9  and  r. 
Q.  E.  D. 


CHAPTER  V. 
THE     PARABOLA. 


Equation  of  the  Parabola. 

97.  Def.  A  parabola  is  the  locus  of  a  jooint  which 
moves  ill  a  plane  in  such  a  way  that  its  distances  from  a  fixed 
point  and  from  a  fixed  straight  line  in  that  plane  are  equal. 

The  fixed  point  is  called  the  focus,  and  the  fixed  straight 
line  the  directrix  of  the  parabola. 

The  curve  is  traced  mechanically  as  follows : 

Let  F  be  the  fixed  point  or  focus,  and  RR'  the  fixed  straight  line  or 
directrix.  Along  the  latter  place  the  edge  of  a 
ruler,  and  to  the  focus  attach  one  end  of  a 
thread  whose  length  is  equal  to  that  of  a  second 
ruler,  DQ,  right-angled  at  D.  Then  having  at- 
tached the  other  end  of  the  thread  to  the  ruler 
at  Q,  stretch  the  thread  tightly  against  the  edge 
of  the  ruler  DQ  with  the  point  of  a  pencil, 
while  the  ruler  is  moved  on  its  edge  BR  along 
the  directrix  RR' :  the  path  of  P  will  be  a 
parabola.      For  in  every  position  we   shall 

have 

PF=PD, 

wiiich  agrees  with  the  definition. 

98.  Problem.    To  find  the  equation  of  tlie  'parabola. 

Let  i^  be  the  focus,  and  YY'  the  directrix.  Through  i^ 
draw  OX  perpendicular  to  YY',  Take  y 
0  as  the  origin,  OF  as  the  axis  of  X, 
and  the  directrix  01^  as  the  axis  of  Y, 
Put  OF  =  p,  and  let  P  be  any  point  on 
the  curve.  Join  PF,  and  draw  PiV^  per- 
pendicular to  the  directrix  YY\  Then, 
by  the  definition  of  the  curve,  we  have 

PF=^PN. 


114 


PLANE  ANALYTIC  OEOMETRY. 


Let  OM,  FM,  the  co-ordinates  of  P,  be  x  and  y.   Then  we 


have 


PM'  +  FM 


that  is, 
or 


y' 


=  FF' 
=  FN' 
=  03P; 

+  {x-py  =  x\ 


y'  =  2p 


_  1 


Ph 


(1) 


which  is  the  equation  of   the  parabola 
with  the  assumed  origin  and  axes. 

When  ?/  =  0  we  have  x  =  ip;  that  is,  OA  =  AF,  or  the 
curve  bisects  the  perpendicular  distance  between  the  focus 
and  the  directrix;  and  since  there  is  no  limit  to  the  possible 
distance  of  a  point  from  both  focus  and  directrix,  the  curve 
extends  out  to  infinity.  From  (1)  we  see  that  for  every  posi- 
tive value  of  X  greater  than  p  there  are  two  values  of  «/,  equal 
in  magnitude  but  of  opposite  signs.  Hence  the  curve  is 
symmetrical  with  respect  to  the  axis  of  X,  If  x  be  negative 
or  less  than  -Jjt?,  the  values  of  y  are  imaginary;  therefore  no 
part  of  the  curve  lies  to  the  left  of  A. 

Def,  The  point  A  where  the  curve  intersects  the  perpen- 
dicular from  the  focus  on  the  directrix  is  called  the  vertex 
and  ^Xthe  axis  of  the  parabola. 

The  equation  (1)  will  assume  a  simpler  and  more  useful 
form  by  transferring  the  origin  to  the  vertex,  which  is  done 
by  simply  writing  x  for  a;  —  -J/?;  hence  (1)  becomes 

f  =  '2px,  (2) 

a  parabola  which  we 
Y 


/ 


which  is  the  form  of  the  equation  of 
shall  use  hereafter. 

In  equation  (2),  let  x  =  ^p. 
Then  y^  =i?% 

or  y  =  ±  p. 

Hence       FL  =  FU  =  2AF, 
and  LU  =  2p     =:  ^AF, 

Def.  The  double  ordinate  through 
the  focus  is  called  the  principal  para- 
meter or  latus  rectum. 

Cor.     The  length  of  the  semi-parameter  is  p. 


L 


THE  PARABOLA. 


115 


99.  Focal  Distance  of  any  Point  on  the  Parabola, 

Let  r  denote  the  focal  distance  FP  of  any  point  P  (§  98). 
Then,  by  the  definition  of  the  curve,  we  have 
FP  =  JVP 

=  OA  +  AM, 
or  r=   i^  +  X,  (3) 

which,  being  of  the  first  degree,  is  sometimes  called  the  lin- 
ear equation  of  the  parabola. 

100.  Polar  Equation  of  the  Parabola. 

Problem.     To  find  the  polar  equation  of  the  parabola, 
the  focus  being  the  pole. 

Let  FP  =  r,  XFP  =  6.  Then,  from  the  figure,  we  have 
FP  =  PJV 

=  OF-\-FM 
=  2AF-{-FM, 
or  r  =p  -{-  r  cos  6; 


whence  r 


P 


1  —  cos  6^ 


2  sin' 


6' 

2 


If  we  count  the  angle  6  from  the 
vertex  in  the  direction  AP,  we  shall 
have  AFP  =  6,  and  therefore  (4)  be- 
comes P         _       P 


1  +  cos  0 


2  cos^  ^ 


(5) 


which  is  the  form  of  the  polar  equation  generally  used. 

Cor.  The  polar  equation  may  also  be  easily  deduced  from 
the  linear  equation  of  the  curve.  Thus,  when  the  vertex  is 
the  origin,  the  linear  equation  is 

r^ip-\-  x; 
and  transferring  the  origin  to  the  focus  by  writing  x  -\-  ^p  for 
X,  it  becomes 

r  =  p  -\-  X 
=  p  —  r  cos  0; 

whence  --  — 

as  before. 


1  +  cos  & 


116 


PLANE  ANALYTIC  OEOMETRT. 


Diameters  of  a  Parabola. 

101,  Def.     The  diameter  of  a  parabola  is  the  locus  of 
the  middle  points  of  any  system  of  parallel  chords. 

Pkoblem.     To  find  the  equation  of  any  diameter. 

Let  X,  y  be  the  co-ordinates  of  P,  the  middle  point  of  any 
chord  CG'\  x' ,  y'  the  co-ordinates  of  C";  r  =  PC,  half  the 
length  of  the  chord;  6  the  inclination  of 
the  chord  to  the  axis  of  the  curve.  Draw 
the  ordinates  PM,  CM'  and  PD  parallel 
to  AM'.     Then  we  have 

AM'  =  AM  ^  PD, 

or  x'  :=i  X  -\-  r  cos  d, 

and         CM'  =  PM-\-C'D, 

or  y^  =  y  -\-  r  sin  d; 

and  since  the   point  (x',  y')  is  on 

curve,  we  have 

y''  =  2px',  {c) 

Substituting  the  values  of  x'  and  y'  as  given  by  (<^)  and 
{h)  in  (c),  we  have 

{y  -j-  r  sin  Oy  =  2p(x  -f  r  cos  0), 
or     r'  sin'^  +  2{y  sin  6  —  p  cos  6)r  +  ^^  —  2px  =  0, 
from  which  tb  determine  the  two  values  of  r.     But  since  the 
point  (x,  y)  is  the  middle  of  the  chord,  the  values  of  r  are 
equal  in  magnitude  but  of  opposite  signs;  therefore  the  co- 
efficient of  r  must  vanish,  which  gives 

y  sin  6  —  p  cos  0  =  0, 
or  y  =  P  cot  6 

-P. 

where  m  =  tan  6,  the  slope  of  the  chord  to  the  axis  of  X, 

Hence  the  equation  of  any  diameter  is 

y  =pcote.  ^  (6) 

Since  the  second  member  of  (6)  is  constant  for  any  system 
of  parallel  chords,  evei^y  diameter  of  a  2Mrabola  is  a  straight 


THE  PARABOLA.  117 

line  parallel  to  the  axis  o/X(§40,  III.).  Because  m  may  have 
any  value  whatever,  (6)  can  be  made  to  represent  any  straight 
line  parallel  to  the  axis  of  the  curve.  Hence  every  line  paral- 
lel to  the  axis  bisects  a  system  of  parallel  chords. 

Cor.  To  draw  a  diameter  of  the  curve,  bisect  any  two 
parallel  chords,  join  the  points  of  bisection  and  produce  the 
line  to  meet  the  curve:  it  will  be  a  diameter. 

Tangents  and  Normals. 

102.  Problem.  To  find  the  equation  of  a  tangent  to  a 
parabola. 

Let  (a;',  ?/')  and  (:c",  y")  be  the  co-ordinates  of  any  two 
points  on  the  curve.  Then  the  equation  of  the  secant  through 
these  points  is 

y-y'  =  P^(^  -  =^')-  («) 

But  since  (a;',  ?/')  and  {a;",  ?/")  are  on  the  curve,  we  have 

«/"  =  2px'  (b) 

and  2/""  =  2i?a;".  (c) 

From  (b)  and  (c)  we  get 

^'-  -  /»  =  2pix^^  -  xy, 

whence  ^„  ~  ^,  =  -^ — ., 

which  substituted  in  (a)  gives,  for  the  equation  of  the  secant, 

y-y'  =  7^^(^  -  ^')- 

Now  when  the  point  (x'^,  ?/")  coincides  with  the  point 
(a;',  y'),  the  secant  will  become  a  tangent,  and  then  a;"  =  a;' 
and  2/"  =  y^;  hence  the  equation  of  the  tangent  at  the  point 
{x',  y')  is 

y  -f  ^  p-(^  -  ^%  0) 

or  y'y  =  i^x  —  p)x^  +  y" 

=  px  —  px*  4-  2px* 
=  P(x  +  x').  (8) 


118  PLANE  ANALYTIC  GEOMETRY. 

Cor.     Let  {x',  y')  be  the  co-ordinates  of  any  point  on  a 
parabola;  the  equation  of  the  tangent  at  that  point  is 

y=P-(x  +  x'),  («) 

and  the  equation  of  the  diameter  passing  through  the  point 
C<  y')  is 

Eliminating  if  from  (a)  and  (J),  we  have 

y  —  m{x  -\-  ic') 
for  the  equation  of  the  tangent.     But  m  is  the  slope  of  the 
parallel  chords  to  the  axis;  hence  the  tangent  at  the  extremity 
of  any  diameter  of  a  parabola  is  parallel  to  the  chords  ivhich 
are  bisected  by  that  diameter. 

The    equation   of  the  tangent  may  also  be  derived  in- 
dependently of  the  point  of  contact  in  the  following  manner. 

103.  PiiOBLEM.     To  find  the  conditio7i  that  the  line 
y  =  mx  -\-  h 
may  be  tangent  to  a  given  parabola. 
The  equation  of  the  curve  is 

whence,  by  eliminating  y  between  these  equations,  we  get 

{mx  -\-hy  =  2px, 
or  nv'x''  +  {27nh  -  2p)x  +  h'  =  0, 

which  determines  the  abscissae  x  of  the  two  points  in  which 
the  line  intersects  the  curve.     But  since  the  line  is  to  be  a 
tangent,  the  two  values  of  the  abscissae  will  be  equal.      The 
condition  that  this  equation  may  have  equal  roots*  is 
{27nh  -  2p)y  =  4.7}fh'; 

whence  h  =  - — , 

2m 

*  The  condition  that  the  roots  of  the  quadratic  equation 
ax"^  +  &a;  +  c  =  0 
shall  be  equal  is  6^  —  iac  =  0.  (Chap.  I.) 


THE  PABABOLA. 


119 


the  required  condition.     Substituting  this  value  of  h  in  the 
equation  of  the  line,  we  have 


y  =  mx  -j-  - 


P 


2m' 


(0) 


which  is  the  equation  of  the  tangent  to  a  parabola  in  terms  of 
the  slope  and  semi-parameter. 

Conversely,  every  line  whose  equation  is  of  this  form  is  a 
tangent  to  a  parabola. 

104.  The  SuUangent.  Def.  The  subtangent  is  the 
projection  of  the  tangent  upon  the  axis  of  the  parabola. 

To  find  where  the  tangent  meets  the  axis  of  X,  make 
2/  =  0,  in  (8),  and  we  get 

0  =  ^(^  +  x'), 
or  .   X  =  —  x'-, 

that  is,     AT=AM', 

or,  the  suUangent  is  bisected  at  the  rp, 
vertex. 

This  property  enables  us  to  draw 
a  tangent  at  any  point  on  a  para- 
bola.    Thus,  let  P  be  any  point  on 
the  curve;  draw  the  ordinate  PM,  and  produce  3IA  to  T, 
making  AT  equal  to  A3f;  join  TP.  Then  TP  is  the  tangent 
required. 

105.  The  Normal.  Def.  The  normal  to  a  curve  at  any 
point  is  the  perpendicular  to  the  tangent  at  that  point. 

Problem.  To  find  the  equation  of  the  normal  to  a  loara- 
tola. 

The  equation  of  the  tangent  at  any  point  {x! ,  y')  has  been 
shown  to  be 


y 


j{x^xy. 


and  let 


y  —  y'  ^z  m{x  —  x') 


(J) 


be  the  equation  of  a  line  through  (a:',  ?/')  and  normal  to  the 
curve  at  that  point. 

Now  in  order  that  the  lines  represented  by  (a)  and  (h) 


120  PLANE  ANALYTIC  OEOMETBT. 

may  be  perpendicular  to  each  other,  we  must  have  the  con- 
dition 

^'  +  1-0,  (§47) 

or  m  =  —  ^; 

P 

therefore  {h)  becomes 

y-y'  =  -j^^-«^')'  (10) 

which  is  the  equation  of  the  normal  at  the  point  (x',  y'). 

106.  The  Subnormal.     Def.     The  subnormal  is  the 

projection  of  the  normal  upon  the  axis  of  the  parabola. 

To  find  where  the  normal  intersects  the  axis  of  X,  make 
2/  =  0  in  (10).     Then  we  have 

p  =1  X  —  x' 
=AN-AM 

that  is,  the  suhnormal  MN  is  coiistant  and  equal  to  half  the 
pammeter  or  latus  rectwn. 

107.  Theorem.  A  tangent  to  a  parabola  is  equally  in- 
clined to  the  axis  of  the  curve  and  the  focal  line  from  the  point 
of  tangency. 

Proof    From  (§  104)  we  have 

FT=  AT-\-  AF 

=  AM-i-  AF 
=  FF; 

therefore  the  angle  PTF  is  equal  to  the  angle  FPT. 

Cor,  Let  PD  be  drawn  parallel  to  the  axis  AX;  then  PD 
is  a  diameter  of  the  curve  (§101),  and  the  angle  IIPI)  is  equal 
to  the  angle  TPF.  Since  the  normal  PJY  is  perpendicular  to 
the  tangent,  the  angle  DPN  is  equal  to  the  angle  NPF, 

Remark.  The  properties  just  proved  find  an  application  in  the  use 
of  parabolic  reflectors  intended  to  bring  rays  of  light  to  a  focus,  as  in 
the  reflecting  telescope.  Since  the  curve  and  the  tangent  have  the  same 
direction  at  the  point  of  tangency,  rays  of  light  are  reflected  by  the 


THE  PARABOLA.  121 

curve  as  they  would  be  by  the  taugeut  at  that  point;  and  because  the 
angle  of  incidence  is  equal  to  the  angle  of  reflection,  it  follows  that  if 
rays  of  light  parallel  to  the  axis  of  the  curve  fall  upon  a  parabolic  reflec- 
tor, they  will  all  be  reflected  to  the  focus.  Conversely,  if  a  luminous 
body  be  placed  in  the  focus  of  a  parabolic  reflector,  all  the  rays  proceed- 
ing therefrom  will  be  parallel  after  reflection. 

108.  Problem.     To  find  the  locus  of  the  foot  of  the  per- 
pendicular from  the  focus  upou  a  variable  tangent. 

Let  x',  if  be  the  co-ordinates  of  any  point  P  on  the  curve. 
The  equation  of  the  tangent  at  P  is 

The  equation  of  the  line  through  the  focus  whose  co-ordi- 
nates are  (^j,  0),  and  perpendicular  to  {ct),  is 

And  since  the  point  {x\  y*)  is  on  the  curve 

y''  =  2px'.  (c) 

we  have  now  to  eliminate  x'  and  y'  from  (a),  (b)  and  (c). 
From  (c), 

x'  -^ 
-2p' 

which  substituted  in  (a)  gives 


From  {b)  we  have 


y'-    py 


X  -  ip 
which  substituted  in  (d)  gives,  after  obvious  reductions, 

{f  +  (^  -  ipy\x  =  0. 

Therefore  we  must  have  either 

y^  +  (x  -  ipy=  0     or     a:  =  0. 

The   former  gives  ^  =  0  and  x  =  ^;,  the   focus,  which 
however  is  not  the  locus  of  the  intersection  of  (a)  and  (b); 


122  PLANE  ANALYTIC  OEOMETRT. 

for  although  these  values  of  x  and  y  satisfy  {b),  they  do  not 
satisfy  {a).     We  conclude,  therefore,  that  the  latter,  namely, 

x^Q,  (11) 

is  the  equation  of  the  required  locus,  which  is  the  tangent  at 
the  origin  or  the  axis  of  Y. 

109.  Pkoblem.  To  find  the  locus  of  the  point  of  inter- 
section of  tivo  ta?igents  to  a  parabola  wJiich  are  perpendicular 
to  each  other. 

Let  the  equation  of  one  of  the  tangents  he 

2/  =  »^  +  |^-  (§103)        {a) 

Then  the  equation  of  the  other,  j^erpendicular  to  (a),  is 

or  my  =  —  x  —  ipni",  (b) 

Multiplying  (a)  by  7n  and  subtracting  (b)  gives 

0  =  2(1  +  'in')x  +  (1  +  m')p, 
or  x=  —  ip,  (12) 

the  equation  of  the  required  locus,  which  is  the  directrix. 

110.  Problem.  To  find  the  length  of  the  perpendicular 
from  the  focus  uyon  the  tangent  at  any  ptoint. 

Let  P  denote  the  length  of  the  perpendicular.  The  equa- 
tion of  the  tangent  at  the  point  {x',  y')  is 

y'y-p{x-\-x')^o,  (§102) 

The  perpendicular  P  from  the  focus,  whose  co-ordinates 
are  (i;;,  0),  is  (§  41) 

^^  p{p-\-'Zx')  ^    p{2x'-}-p) 
2Vjr-\-p^       2V2px'  -{-p' 
=  i  i^pip  +  3a;') 
=  ^V2pr,  (§99)         (13) 

where  r  is  the  focal  distance  of  the  point  of  tangency. 


THE  PARABOLA. 


123 


111.  Problem.  To  find  the  co-ordinates  of  the  point  of 
contact  of  a  tangent  drawn  from  a  given  point  to  a  parabola. 

Let  {x',  y')  be  the  co-ordinates  of  the  required  point  of 
contact,  and  {h,  k)  the  co-ordinates  of  the  given  fixed  point. 
The  equation  of  the  tangent  at  (x',  y')  is 

and  since  the  tangent  passes  through  the  point  (h,  h),  we  also 
have 

ky'=p{x'-\-h),  {a) 


Because  the  point  {x',  y')  is  on  the  curve,  we  have 


y'-  —  "Ipx' 


(S) 


Solving  (ci)  and  (^)  for  x'  and  y* ,  we  have 
x'  =  F  -  ^h  ±  h  \/Tc 


2ph', 


These  equations  show  that  from  any  fixed  point  two  tan- 
gents can  be  drawn  to  a  parabola,  and  that  the  points  of  con- 
tact {x\  y')  will  be  real,  coincident  or  imaginary  according 
as  k"^  —  2/)/i  >  0,  =0,  or  <  0;  that  is,  according  as  the 
point  {h,  k)  is  ivithout,  on  or  icithin  the  curve. 

11^.  Problem.  To  find  the  equation  of  the  paralola 
referred  to  any  diameter  and  the  tangent  at  its  vertex,  as  axes. 

Let  A'  be  any  point  on  a 
parabola;  take  this  point  as 
origin  and  draw  through  it 
the  diameter  A'X'  for  the 
new  axis  of  X,  and  the  tangent 
TA'Y'  for  the  new  axis  of  Y. 

Let  Y'A'X'=A'TX=e, 
and  h  and  k  the  co-ordinates 
of  A'  referred  to  the  original 
axes  AXy  AY. 

Let  ix,  y)  be  the  co-ordi- 
nates of  any  point  P  referred  to  the  original  axes,  and  (re',  y') 
the  co-ordinates  of  the  same  point  referred  to  the  new  axes; 
draw  the  ordinates  PM,  PM' ,  and  draw  if'iV^and-4'^paral- 


124  PLANE  ANALYTIC  GEOMETRY. 

lei  to  A  Y,  and  let  Q  denote  the  intersection  of  the  diameter 
A'X'  and  the  ordinate  FM.    Then,  from  the  figure,  we  have 

x  =  AM=  AH-i-  A'M'  +  M'Q 

=  h^x'  -{-  PM'  cos  PM'Q 
=  h  -\-  x'  -{-  y'  cos  dy  (a) 

and  y  =  PM=A'H+PQ 

=  ]c-\-  PM'  sin  PM'Q 

=  k  ^  y'  sin  0.  {b) 

But  since  the  point  {x,  y)  is  on  the  curve, 

y'  =  2px.  (c) 

Substituting  the  values  of  x  and  y  as  given  by  (a)  and  (b) 
in  (c),  we  have 

(k  4-  2/'  sin  ey  =  2p(h  +  x'  +  y'  cos  ^); 
whence 

y"  sin'  0  +  22/'(^  sin  6^  -i?  cos  6)-^  ¥  -  2jjh  =  22)x\ 

But,  by  (6)  of  §101, 

Jc  =p  cot  6; 

and  since  the  point  {h,  Jc)  is  on  the  curve 

F  =  2ph, 
therefore  we  have 

y'^  sin'  0  =  2px' , 

which  is  the  equation  of  the  curve  referred  to  the  new  axes. 

Cor.     This  equation  may  also  be  expressed  in  terms  of 
A'F,  the  focal  distance  of  the  point  A\    Thus,  by  (3)  of  §99, 

A'F=ip-\-h, 

k* 
and  k^  =  2ph,        or        h  =  ^. 

k^ 
Therefore  A'F=ip-\--- 

^p 

=  i{p-\-pooi^e) 

(since  k=p  cot  6) 
=  ip(l  4-  cot'  d) 

^      P 
2  sin'  6' 


THE  PARABOLA.  125 

Therefore,  denoting  A'Fhj  ^p',  equation  (14)  may  be  written 

or,  suppressing  the  accents  on  the  variables, 

y'  =  2p'x,  (15) 

Cor.     From  the  identity  of  form  in  the  equations 

^'  =  2px        and        y""  =  )lp'Xy 

we  may  at  once  infer  that  the  equation  of  the  tangent  referred 
to  any  diameter  is 

y'y=p'{x-^x').  (16) 

If  in  this  equation  we  put  j^  =  0,  we  get 

X     ^=      —     X''y 

or,  the  intercept  on  the  axis  of  X  is  equal  to  the  abscissa  of 
the  point  of  contact,  and  therefore  the  subtangent  to  any 
diameter  of  a  parabola  is  bisected  by  the  vertex. 

Poles  and  Polars. 

113.  Def.  A  chord  of  contact  is  the  chord  joining 
the  points  of  contact  of  two  tangents. 

Problem.  To  find  the  equatio7i  of  the  chord  of  contact 
of  tiuo  tangents  from  an  external  point. 

Let  (ojj,  yj  be  the  co-ordinates  of  the  external  point, 
(a;',  y')  the  co-ordinates  of  the  point  where  one  of  the  tangents 
through  (x^j  y^  meets  the  curve,  and  {x",  y")  the  co-ordi- 
nates of  the  point  where  the  other  tangent  meets  the  curve. 

The  equation  of  the  tangent  at  (x' ,  y')  is 

ify=p{x'  -^x);  ^  (a) 

and  since  this  passes  through  (x^,  ?/j),  we  have 

and,  for  the  same  reason, 

yy  =  p(x''  -}-  x;).  (c) 

Hence  the  equation  of  the  chord  of  contact  is 

y^y  =p{x-{-x;),  (17) 


126  PLANE  ANALYTIC  GEOMETRY. 

for  this  is  the  equation  of  a  straight  line,  and  is  satisfied  by 

X  =  x\        y  =  y'        and        x  =  x" ,        y  —  y", 

as  we  see  from  (h)  and  (c).  Therefore  (17)  is  the  equation  of 
the  chord  of  contact  of  the  tangents  through  the  poi^it  {x^,  y^, 

114,  Locus  of  the  Poiiit  of  Intersection  of  Two  Tangents. 

Let  {x^,  y^  be  the  co-ordinates  of  any  fixed  point  through 
which  a  chord  of  contact  of  two  intersecting  tangents  is  drawn, 
and  (a;^,  i/J  the  co-ordinates  of  the  point  of  intersection  of  the 
taugents.  Then  the  equation  of  the  chord  of  contact  is,  by 
the  last  section, 

y^y  =i?(^\  +  ^); 

but  since  {x^,  y^  is  a  point  on  the  chord,  we  must  also  have 
the  condition 

y^y,  =  PC^\  +  ^^)y 

which  the  co-ordinates  of  the  point  of  intersection  must  al- 
ways satisfy,  however  the  chord  of  contact  may  change  its 
position  as  it  revolves  about  the  fixed  point  (x^^  y^.  Therefore 
the  equation  of  the  required  locus  is 

y^y  =  p{^  +  ^^)y  (IS) 

which  is  that  of  a  straight  line.     Hence  we  have  the  theorem: 

If  through  any  fixed  point  tve  draw  chords  to  a  parabola; 

and  if  through  the  ends  of  each  chord  we  draiu  a  pair  of 
tangents, 

then  the  point  of  meeting  of  every  pair  of  tangents  will  lie 
on  a  certain  straight  line. 

Def  Such  straight  line  is  called  the  polar  of  the  point 
through  which  the  chords  pass. 

It  follows  from  this  theorem  that  if  (x^,  y^)  be  any  fixed 
point,  the  equation  of  the  polar  of  that  point  is 

y,y  =  p)(x  +  x^)  (19) 

when  referred  to  the  axis,  or 

y.y  =  p\^  +  ^.)  (^0) 

if  referred  to  a  diameter  and  a  tangent  at  its  vertex  as  axes. 


THE  PARABOLA.  127 

Direction  of  the  Polar. 
Making  y^  =  0  in  (20),  we  have 

x=  ~  a;,,  (21) 

which  is  the  equation  of  a  line  parallel  to  the  axis  of  Y.    Hence 
The  polar  of  any  point  is  parallel  to  the  tangent  at  the  end  of 

the  diameter  ptcLssing  through  that  pointy  and  is  situated  at  a 

distance  from  the  vertex  of  the  diameter  equal,  but  in  an  optpo- 

site  direction,  to  the  distance  of  the  point. 
115.  Polar  of  tlie  Focus. 
Put  {^p,  0)  for  x^,  y^  in  the  equation  of  the  polar,  and  we 

get 

x=  -  ip,  (22) 

which  is  the  equation  of  the  directrix.     Therefore 
Tlie  polar  of  the  focus  of  a  parabola  is  the  directrix. 

EXERCISES. 

1.  Find  the  points  of  intersection  of  the  line  ?/  =  3a;  —  6 
with  the  parabola  y"^  —  9:c.  Ans.  (4,  6)  and  (1,—  3). 

2.  Find  the  equation  of  a  line  through  the  focus  of  the 

parabola  ?/^  =  12rc  and  making  an  angle  of  30°  with  the  axis 

of  iC.  .  X  ^ 

^t^8.  y  =  —-^, 

3.  Find  the  equation  of  the  line  through  the  vertex  and 
the  extremity  of  the  latus  rectum.  Ans.  y  =  ±  ^x. 

4.  Find  the  equation  of  the  circle  which  passes  through 
the  vertex  of  a  parabola  and  the  extremities  of  the  latus  rec- 
tum. Ans.  x^  -\-  y^  =  ^px. 

5.  Find  the  equation  of  the  tangent  at  the  extremity  of 
the  latus  rectum,  and  the  angle  between  this  tangent  and 
the  line  drawn  to  the  vertex  from  the  same  extremity  of  the 
latus  rectum.  Ans.  y  =  x  -{-  ^p;    tan^~^^i. 

6.  Determine  the  equations  of  the  normals  at  the  extrem- 
ities of  the  latus  rectum,  the  co-ordinates  of  the  points  in 
which  these  normals  again  intersect  the  curve,  and  the  length 
of  the  chords  formed  by  the  normals. 


128  PLANE  ANALYTIC  GEOMETRY. 

7.  Show  that  if  the  focus  of  a  parabola  is  the  origin,  and 
the  axis  of  the  curve  the  axis  of  X,  the  equation  of  the  para- 
bola is  y^  =  p(2x  +  jy),  and  the  equation  of  the  tangent  at 
the  point  {x\  y')  is 

y'y  =  p{^  +  ^'  +i?). 

8.  With  the  same  origin  and  axes  as  in  the  last  example 
show  that  the  equations  of  the  tangents  and  normals  at  the 
extremity  of  the  latus  rectum  are 

X  T  y  -\-  p  =  0; 
X  ±  y  —  p  =  0, 

9.  Prove  that  the  circle  described  on  any  focal  chord  as 
diameter  will  touch  the  directrix. 

10.  A  tangent  is  drawn  to  a  parabola  at  the  point  (.^•',  ?/'). 
Find  the  length  of  the  perpendicular  drawn  from  the  foot  of 
the  directrix  on  this  tangent. 

Ans.      ^         ^ 


)lVy'^-\-p^ 

11.  Pairs  of  tangents  are  drawn  to  a  parabola  at  points 
whose  abscissas  are  in  a  constant  ratio.  Show  that  the  locus 
of  the  intersection  of  the  tangents  is  a  parabola. 

12.  Find  the  polar  equation  of  the  parabola  when  the 
vertex  is  the  pole,  and  the  axis  of  the  curve  the  initial  line. 

Alls,  r  =  2p  cot  6  cosec  6. 

13.  If  r  and  r'  be  the  lengths  of  two  radii  vectores  drawn 
at  right  angles  to  each  other  from  the  vertex  of  a  parabola, 
show  that 

14.  Find  the  equation  of  the  parabola  referred  to  the  tan- 
gents at  the  extremities  of  the  latus  rectum  as  axes. 

Ans.  {x  —  yY  —  2  V2p{x  +  y)  -^2p'  =  0. 

15.  If  tangents  be  drawn  to  a  parabola  at  the  extremities 
of  any  focal  chord,  show  that  they  will  intersect  at  right 
angles  on  the  directrix,  and  that  the  line  from  their  point  of 
intersection  to  the  focus  is  perpendicular  to  the  focal  chord. 


'Jr/r- 


THE  PARABOLA.  129 

16.  From  an  external  point  {x\  if)  two  tangents  are  drawn 
to  a  parabola.   Show  that  the  length  of  the  chord  of  contact  is 


2(ir 

J^f)\y'^-2px')^ 

P 

3 

and  that  the 

area 

of  the 

J  triangle  formed 

by  the  chord  and  tan- 

gents  is 

P 

17.  If  ni,  m'  be  the  slopes  to  the  axis  of  the  parabola  of 
the  two  tangents  in  the  last  example,  show  that 

m-\-m'  =  ~        and        mm'  =  r^. 

X  n/X 

18.  If  {x',  y')  and  (re",  y")  be  any  two  points  on  a  para- 
bola, show  that  the  tangent  of  the  angle  contained  by  the 
tangents  touching  at  these  points  is 

p{y"  -  y') 
f  +  y"y' 

19.  In  what  ratio  does  the  focus  of  a  parabola  divide  that 
segment  of  the  axis  cut  out  by  a  tangent  and  normal  drawn 
at  the  same  point  of  the  parabola? 

20.  A  triangle  is  formed  by  three  tangents  to  a  parabola. 
Show  that  the  circle  which  circumscribes  this  triangle  passes 
through  the  focus. 

21.  Show  that  the  parameter  of  any  diameter  is  equal  to 
four  times  the  focal  distance  of  its  vertex,  or  equal  to  the  focal 
double  ordinate  of  that  diameter. 

Note.     The  parameter  of  any  diameter  is  the  focal  chord  bisected 

by  that  diameter,  called  2y  in  §  112. 

I 

22.  If  TP  and  TQ  are  tangents  to  a  parabola  at  the  points 
P  and  Qi  then  if  F  be  the  focus,  show  that 

FP  .  FQ=  FT\ 


130  PLANE  ANALYTIC  OEOMETRT. 

23.  Sliow  that  tlie  area  of  the  triangle  in  Prob.  20  is  half 
that  of  the  triangle  formed  by  joining  the  points  of  contact 
of  the  three  tangents. 

24.  Given  the  outline  of  a  parabola,  show  how  to  find  the 
focus  and  the  axis. 

25.  The  base  of  a  triangle  is  2a,  and  the  sum  of  the  tan- 
gents of  the  base-angles  is  m.  Show  that  the  locus  of  the  ver- 
tex is  a  parabola  whose  semi-parameter  is  — . 

26.  Prove  that  y  —  x  tan  0  +jt?  cosec  20  is  a  tangent  to 
a  parabola  whose  latus  rectum  is  p,  the  origin  being  at  the 
focus,  and  the  axis  of  the  curve  the  axis  of  X. 

27.  Tangents  are  drawn  from  any  two  points  P,  §  to  a 
parabola.  Show  that  the  co-ordinates  of  T,  the  intersection 
of  the  tangents,  are 

1    cos  {B,  +  e,)         \    sin  ((9,  -f  (9,) 
4^  sin  0,  sin  0/        4^^  sin  0,  sin  0/ 

where  tan  Q^  and  tan  6^  are  the  slopes  of  the  tangents  to  the 
axis  of  X, 

28.  If  all  the  ordinates  of  a  parabola  are  increased  in  the 
same  ratio,  show  that  tlie  new  curve  will  be  a  parabola,  and 
express  its  parameter  in  terms  of  the  ratio  of  increase. 

29.  At  what  point  of  a  parabola  is  the  normal  double  the 
subtangent;  and  what  angle  does  that  normal  form  with  the 
axis  of  the  parabola? 

30.  Find  a  point  upon  a  parabola  such  that  the  rectangle 
contained  by  the  tangent  and  normal  shall  be  twice  the  square 
of  the  ordinate;  and  show  the  relation  of  such  point  to  the 
focus. 

31.  Find  that  point  on  a  parabola  for  which  the  normal  is 
equal  to  the  difference  between  the  subtangent  and  the  sub- 
normal. 

32.  Having  given  the  parabola  if  —  6a',  find  the  equation 
of  that  chord  wliich  is  bisected  by  the  point  (4,  3). 

33.  Find  the  equation  of  that  chord  of  a  parabola  which 
is  drawn  from  the  vertex  and  bisected  by  the  diameter  y  —  q* 


CHAPTER  VI. 
TH  E     ELLI  PSE 


Equations  and  Fundamental  Properties. 

116.  Def.  An  ellipse  is  the  locus  of  a  point  the  sum 
of  whose  distances  from  two  fixed  points  is  constant. 

The  two  fixed  points  are  called  foci  of  the  ellipse.  Thus, 
if  the  point  P  move  in  such  a  way  that  PF -\-  PF'  is  con- 
stant, it  will  describe  an  ellipse. 

The  curve  may  be  described  me- 
chanically as  follows:  Take  any  two 
fixed  points  i^'and  F' ,  and  attach  to 
tbera  tbe  extremities  of  a  thread  whose  £ 
length  is  greater  than  the  distance  FF' . 
Place  a  pencil-point  P  against  the 
thread,  and  slide  it  so  as  to  keep  the 
thread  constantly  stretched:  the  point 
P  will  describe  an  ellipse,  for  in  every 
position  we  shall  have  PF  -\-  PF'  =  the  constant  length  of  the  thread. 

The  line  AA^  drawn  through  the  foci  and  terminated  by 
the  curve  is  called  the  transverse  or  major  axis,  and  BB' 
bisecting  AA'  at  right  angles  is  called  the  conjugate  or 
minor  axis.    The  two  are  called  principal  axes. 

The  semi-axes  CA  and  CB  are  represented  by  the  symbols 
a  and  i  respectively. 

The  point  C  midway  between 
the  foci  is  called  the  centre. 

From  the  manner  in  which  the 
curve  is  generated,  we  see  that       A I 

AF=  A'F' 
and 

PF-\.  PF'  =  AA'. 


132  PLANE  ANALYTIC  GEOMETRY. 

117.     Problem.      To  find  the  equation  of  the  ellipse. 

Solution.  Let  C,  the  intersection  of.  AA^  and  BB^,  be 
the  origin;  CA  the  axis  of  X,  and  GB  the  axis  of  Y;  put 
CA  =  CA'  =  a,  CF  —  OF'  =  c,  and  x,  y  the  co-ordinates 
of  any  point  P  on  the  locus.     Then  we  shall  have 


PF   1=  VPM'  +  MF'    =  Vif  +  (c  -  xY; 
PF'  =  VPM'  +  MF''  =  Vy'  +  (c  +  x)\ 
Therefore,  by  definition. 


Clearing  this  equation  of  surds,  it  reduces  to 

(«'  -  c')x'  +  ay  =  a\a'  -  c'). 
But,  by  definition, 

a'  -c'  =  BF'  -  CF'  =  BC  =  b'; 
therefore  we  have,  by  substituting  in  the  above, 

i^x'  +  a'y'  =  a-F; 
or,  dividing  by  a'^b',  we  have 


X'    .    y 


a' 


+  f.  =  1,  (1) 


which  is  the  simplest  form  of  the  equation  of  the  ellipse.  It 
is  called  the  equation  of  the  ellipse  referred  to  its  centre  and 
axes,  because  the  centre  is  the  origin  and  the  axes  are  the 
axes  of  co-ordinates. 

Def.     The   distance  CF  =  CF'  =  c  between  the  centre 
and  either  focus  is  the  linear  eccentricity  of  the  ellipse. 

The  ratio  —  of  the  linear  eccentricity  to  the  semi-major 

axis  is  called  the  eccentricity  of  the  ellipse. 
By  the  common  notation, 

a  a  ^  ' 

is  the  expression  for  the  eccentricity  in  terms  of  the  semi- 
axes. 


THE  ELLIPSE.  133 

Oor,  If  we  transfer  the  origin  to  A',  whose  co-ordinates 
are  (—  a,  0),  the  equation  (1)  becomes,  by  writing  {x  —  a) 
for  Xy 

(^  -  ctY  ,  .1'  _  1 

or  y'"^a'  ^^^^  ~  ''^')' 

a  form  of  the  equation  of  the  ellipse  which  is  sometimes  use- 
ful. 

EXERCISES. 

1.  Find  the  eccentricity  and  semi-axes  of  the  ellipse 

16x'  +  26  f  =  400. 

Remark.  Reduce  the  second  member  to  unity  by  dividing  by  400, 
and  compare  with  (1). 

2.  What  are  the  semi-axes  and  the  equation  of  the  ellipse 
when  the  distance  between  the  foci  is  2  and  the  sum  of  the 
distances  from  each  point  of  the  curve  to  the  foci  is  4? 

3.  Determine  the  eccentricity  and  semi-axes  of  the  ellipses 
having  the  following  equations: 

(a)     x'  +  2y'  =  6;    (b)  3x'  +  4/  =  9;    (c)  4a:'  +  9i/  =  16; 
(d)  mx'  +  nf  =p;    (e)  ^x^  +  1^  =  ^    if)  «^'  +  if  =  L 

4.  Using  the  preceding  notation,  prove  the  following  pro- 
positions: 

I.  The  distance  of  either  focus  from  the  centre  is  ae. 

II.  The  distance  of  either  focus  from  the  nearest  end  of 
the  major  axis  is  a{l  —  e). 

III.  The  distance  of  either  focus  from  the  farther  end  of 
the  major  axis  is  a(l  -f-  e). 

IV.  The  distance  from  either  end  of  the  major  axis  to 
either  end  of  the  minor  axis  is  a  V2  —  e^. 

V.  If  we  define  an  angle  cp  by  the  equation 

sin  q)  =  e, 
we  shall  have  for  the  semi-minor  axis 
h  =  a  cos  q}. 


134 


PLANE  ANALYTIC  OEOMETRY. 


5.  Find  the  points  in  which  the  circle  x"  -\-  y^  =  ^  inter- 
sects the  ellipse  x""  -f  ^if  —  G. 

6.  Write  the  equation  of  that  ellipse  whose  minor  axis  is 
10  and  the  distance  between  whose  foci  is  12. 

118.  If  we  solve  equation  (1)  with  respect  to  y,  we  find 

h 


y 


±  -  Va'  -  x\ 
a 


This  equation  shows  that  for  every  value  of  x  there  will 
be  two  values  of  y,  equal  but  with  opposite  signs.  Hence  the 
curve  is  symmetrical  with  respect  to  the  major  axis. 

By  solving  with  respect  to  x  we  show  in  like  manner  that 
the  curve  is  symmetrical  tvith  respect  to  the  7ninor  axis. 

Def.  A  chord  of  an  ellipse  is  any  straight  line  terminated 
by  two  points  of  the  ellipse. 

A  diameter  of  an  ellipse  is  any  chord  through  its  centre. 

Cor.     The  major  and  minor  axes  are  diameters. 

Def.  The  parameter  or  latus  rectum  is  a  chord 
through  the  focus  and  perpendicular  to  the  major  axis. 

119.  Theorem  I.  The  parameter  of  an  ellipse  is  a  third 
proportiojial  to  the  major  and  minor  axes. 

Proof.  The  semi-parameter  is,  by  definition,  the  value  of 
the  ordinate  y  when  x  =  ae.     From  equation  (1),  we  have 


x^). 


tion  in  this  equation. 


P 

Hence  p 


a- 

~  a 
a  :  l 


(«» 


If  we  put  p  for  the  semi-parameter,  we  find,  by  substitu 

h^ 
a'' 

or  a  :  0  =  h  :  p. 

Cor.     The  length  of  the  semi-parameter  FL  is 
p  =  a{l-  e'). 


or 


ap  = 


(3) 


TEE  ELLIPSE. 


135 


120.  Focal  Radii,  or  Radii  Vedores. 

Def.     The  focal  radii  of  an  ellipse  are  the  lines  drawn 

from  any  point  on  the  curve  to  the  foci. 

Problem.  To  express  the  lengths 
of  the  focal  radii  in  terms  of  the  ab- 
scissa of  the  point  from  which  they 
are  draion. 

Let  r  and  r'  be  the  focal  radii  of 
the  point  P,  whose  co-ordinates  are 

{^^  y)' 

Becanse  FG  =  OF'  =  ae, 
we  have     r'  =  FM'  +  PM' 
=  (x  —  ae)'  +  y^ 

=  {X  -  aey  +  ^,  {a^  -  x^) 

=  x'  -  2aex  +  a'e'  +  (1  -  e'){a'  -  x") 
=  a^  —  2aex  -j-  e'^x^. 
Therefore  r  =  a  —  ex.  (4) 

In  the  same  way  we  find,  for  the  other  focal  radius, 

7*'  =  rt  +  ex.  (5) 

These  expressions  are  of  remarkable  simplicity,  and,  being 
of  one  dimension  in  x,  either  of  them  is  called  the  linear 
equation  of  the  ellipse. 

We  observe  that  their  sum  is  2a,  as  it  should  be. 

Cor.  Equations  (4)  and  (5)  show  that  if  a  point  move  on 
the  circumference  of  an  ellipse  in  such  a  way  that  its  abscissa 
increases  unifoi^mly,  one  focal  radius  tvill  increase  and  the 
other  will  decrease  uniformly. 

In  other  words,  if  the  abscisses  of  several  points  are  in 
arithmetical  jjrogi'ession,  their  focal  radii  ivill  also  be  in 
arithmetical  progression. 

121.  Polar  Equation  of  the 
Ellipse,  the  right-hand  focus  being 
the  pole. 

Let  r  and  6  be  the  polar  co- 
ordinates of  any  point  P  on  an 
ellipse;    that    is,  r  =   FP    and 


136  PLANE  ANALYTIC  GEOMETRY. 

6  =  the  angle  AFP.     Join  PF'.     Then,  from  the  triangle 
FPF',  we  have 

PF""  =  PF'  4-  FF"  -  2PF.  FF  .  cos  PFF, 
But  FF'  =  2ac        and        cos  PFF'  =  -  cos  AFP; 


therefore    PF'  =  Vr'  +  Aa'e'  +  4.aer  cos  6, 
and  by  the  fundamental  property  of  the  ellipse  we  have 
PF-\-  PF'  =  A  A', 


or  r  +  V?-""  +  4a'e^  -f  4aer  cos  6  =  2a; 

whence  we  easily  find 

which  is  the  required  equation. 

The  polar  equation  may  also  be  easily  obtained  from  the 
linear  equation  of  the  ellipse;  thus,  from  (4),  we  have 

r  =  a  —  ex, 

the  origin  being  at  the  centre. 

Transferring  the  origin  to  the  right-hand  focus,  whose  co- 
ordinates are  (ae,  0),  it  becomes 

r  =  a(l  —  e"^)  —  ex, 

which  in  polar  co-ordinates  becomes 

r  =  a(l  —  e"^)  —  er  cos  6; 

whence  r  =  zr-^. ^-, 

1  +  e  cos  t^ 

as  before. 

If  the  left-hand  focus  be  taken  as  the  pole,  the  student 
may  easily  show  thiit  the  polar  equation  is 

a(l  -  e-") 
1  —  e  cos  6' 

Cor.    It  0  =  0,  we  have  r  =  ^\    .    ^  -  =  a(l  —  e),  which 
is  the  value  of  AF. 


THE  ELLIPSE.  137 

When  6  =  180°,  we  get  r  =  a(l  +  e),  which  is  the  value 
of  A'F. 

When  6  =  90°,  r  =  a(l  —  e^),  the  semi-parameter. 
These  results  agree  with  those  of  §§  117,  119. 

EXERCISES. 

1.  If  the  semi-minor  axis  of  an  ellipse  is  b,  and  the  eccen- 
tricity sin  cpj  express  its  semi-major  axis  and  semi-parameter 
in  terms  of  b  and  cp.  Ans.  a^=h  sec  q)\ 

p  z=  h  cos  <p. 

2.  The  distance  from  the  focus  to  the  nearer  end  of  the 
major  axis  is  2,  and  the  semi-parameter  is  3.  Find  the 
major  and  minor  axes  and  the  eccentricity. 

3.  Express  the  ratio  of  the  parameter  to  the  distance  be- 
tween the  focus  and  either  end  of  the  major  axis. 

4.  The  major  axis  is  divided  by  the  focus  into  two  seg- 
ments. Show  that  the  rectangle  contained  by  these  segments 
is  equal  to  the  rectangle  contained  by  the  semi-major  axis  and 
the  semi-parameter,  and  also  equal  to  the  square  of  the  semi- 
minor  axis. 

5.  Write  the  equation  of  an  ellipse  in  terms  of  its  semi- 
minor  axis  h,  and  its  semi- parameter  j(?. 

Ans.  fx^  4-  yy''  =  ^"' 

6.  Find  the  points  in  which  the  several  straight  lines 

y  =  %x,         y^lx^l,        y  =  2x  +  2, 

intersect  the  ellipse  x'^  +  2y^  =  6,  and  the  lengths  of  the  three 
chords  which  the  ellipse  cuts  out  from  the  lines. 

7.  Find  the  equation  of  the  ellipse  when  the  right-hand 
focus  is  the  origin,  the  axes  being  the  major  axis  and  the 
latus  rectum. 

x'^   ,   w"*   ,   2ex       F 
Ans.  -o  +  fo  H =  -o. 

8.  The  sum  of  the  principal  axes  of  an  ellipse  is  108,  and 
the  linear  eccentricity  3G.  Find  the  equation  of  the  ellipse, 
and  the  eccentricity. 

Ans.  39". +  15^  =  1;    ^  "^  13* 


138  PLANE  ANALYTIC  OEOMETRT. 


Diameters  of  an  Ellipse. 

122.  Theorem  II.  Every  diameter  of  an  ellipse  u  hi- 
sected  hy  the  centre. 

Proof.  Let  y  =  mxhe  the  equation  of  any  line  through 
the  centre.  Eliminating  y  between  this  equation  and  that  of 
the  ellipse,  we  have 

a''^~¥~~ 

from  which  to  determine  the  abscissae  of  the  points  in  which 
the  line  intersects  the  ellipse.  Since  this  equation  contains 
terms  in  x^  but  none  in  x,  it  will  reduce  to  a  pure  quadratic, 
of  which  the  two  roots  are  equal  but  with  opposite  signs. 
From  these  roots  we  shall  get,  by  substituting  in  the  equation 
y  =  mx,  two  equal  values  of  y  with  opposite  signs.  Hence 
the  points  of  intersection  are  at  equal  distances  on  each  side 
of  the  origin. 

123.  Theorem  III.  The  locus  of  the  centres  of  parallel 
chords  of  an  ellipse  is  a  diameter. 

Proof.  Let  y  =  mx -{- h  (a) 
be  the  equation  of  a  chord;  the 
slope  m  being  the  same  for  all 
the  chords,  while  h  varies  from 
one  chord  to  another. 

We  first  find  the  points  of  in- 
tersection of  the  chord  with  the  ellipse  in  the  usual  way. 

Eliminating  2/ between  (a)  and  the  equation  of  the  ellipse, 
we  find  the  abscissae  of  the  points  of  intersection  to  be  deter- 
mined by  the  quadratic  equation 

x^      (mx  -f  hY  _ 


which  being  reduced  to  the  general  form  becomes 

_2aM_        a\h'-  b')  _ 
""  +  a'm'  -f  Z^^^  "^  a'm''  +  I' 

Now  we  need  not  actually  solve  this  equation  to  obtain 


THE  ELLIPSE.  139 

the  result  we  want,  namely,  the  abscissa  of  the  middle  point 
of  the  chord.     We  know  that  if  we  put,  for  brevity, 

^  ~  a'm'  +  b' ' 
and  call  the  roots  x^  and  x^,  we  shall  have 


which  give  the  abscissas  of  the  two  points  in  which  the  chord 
intersects  the  ellipse.     The  corresponding  values  of  y,  from 

(a),  are 

y^  z=z  mx^  +  h; 

y^  =  mx,  +  h. 

By  (§23),  the  co-ordinates  of  the  middle  point  of  the 
chord  are  the  half -sums  of  the  co-ordinates  of  the  extremities. 
If,  then,  we  put  x\  y'  for  the  co-ordinates  of  the  middle  point 
of  the  chord,  we  have 

,  _       ^    _  ct^mh 

y'  =  im{x^  +  x^)  +  h 
=  rnx'  -\-  h 

bVi 


The  problem  now  is,  What  relation  exists  between  x'  and 
y'  when  we  suppose  h  to  vary  and  all  the  other  quantities  which 
enter  the  second  member  of  (b)  to  remain  constant?  We  ob- 
tain this  relation  by  eliminating  h  between  the  two  equations, 
which  is  done  by  multiplying  the  first  by  b^  and  the  second 
by  a'm  and  adding  the  products.     Thus  we  find 

b'x'  +  a'm/^O.   ^  (7) 

This  is  a  relation  between  the  co-ordinates  of  the  middle 
points  of  the  parallel  chords  which  is  true  for  all  values  of  h, 


140  PLANE  ANALYTIC  GEOMETRY. 

that  is,  for  nil  such  chords;  it  is  therefore  the  equation  of 
the  required  locus,  and,  from  its  form,  is  a  straight  line 
through  the  origin  and  therefore  through  the  centre  of  tlie 
ellipse. 

124.   Conjugate  Diameters.     If  we  omit  the  accents  in 
(7),  we  may  write  it  in  the  form 

^  am 

By  assigning  different  values  to  m,  or,  which  is  the  same 
thing,  by  giving  different  directions  to  the  parallel  chords,  the 

slope ^-  may  take  all  possible  values,  and  therefore  (7) 

may  represent  any  line  passing  through  the  centre  and  bisect- 
ing a  system  of  parallel  chords. 

If  m'  be  the  slope  of  the  diameter  which  bisects  all  the 
chords  whose  slope  is  m,  we  have 

y  z=  m'x, 

the  equation  of  the  diameter; 

but,  by  (7),  y  = =—  x 

•^  ^  ^  ^  a^m 

is  also  the  equation  of  the  diameter. 

Therefore  m*  = 5--, 

am 

or  mm'  = 5-.  (8) 

a  ^  ' 

Theorem  IV.  If  one  diameter  bisects  chords  parallel  to 
a  second  diameter,  the  second  dia^neter  will  bisect  all  chords 
parallel  to  the  first. 

Proof.  If  m  and  m'  be  the  respective  slopes  of  the  two 
diameters,  we  shall  have 

^' 

mm   = 5-, 

a 

since  the  first  bisects  all  chords  parallel  to  the  second;  but 
this  is  also  the  only  condition  which  must  hold  in  order  that 
the  second  may  bisect  all  chords  parallel  to  the  first. 


THE  ELLIPSE. 


141 


Def.  Two  diameters  each  of  which  bisects  all  chords  par- 
allel to  the  other  are  called  conjugate  diameters. 

Cor.  As  the  chords  of  a  set  become  indefinitely  short  near 
the  terminus  of  the  bisecting  diameter,  they  coincide  in  direc- 
tion with  the  tangent  at  the  terminus.     Hence: 

Theorem  V.  The  tangent  to  an  ellipse  at  the  end  of  a 
diameter  is  parallel  to  the  conjugate  diameter. 

125,  Problem.  Given  the  co-ordinates  of  the  extremity 
of  one  diameter,  to  find  those  of  either  extremity  of  the  con- 
jugate diameter. 

Solution.     Let  P CP'  and  D CD'        p^-- --^  p 

be  any  pair  of  conjugate  diameters, 
and  (x',  y')  the  given  co-ordinates 
of  P.     Then  the  equation  of  GP  is 


y 

-^Lx 

(since  m 

-yL\ 

~  xn 

and  the 

equation 

of  DD'  is 

y  =  - 

am 

or 

y^- 

¥x' 

and  the  equation  of  the  ellipse, 

aY  +  2''^'  =  a'^'. 
Substituting  from  {b)  in  (c),  we  have 

{.Vx'-"  +  a'y''')x^  =  aY'; 
but  since  («',  ?/')  is  on  the  ellipse,  we  have 
b^x''  +  a'y''  =  a'b'; 
therefore  a^'b^x''  =  a*y'^y 


(x,y) 


(0) 


or 


X=±p' 


Substituting  this  value  of  x  in  (a),  we  get 


^  a 


142 


PLANE  ANALYTIC  QEOMETRT. 


136.  Theorem  VI.  The  sum  of  the  squares  of  tioo  con- 
jugate semi-diametei^s  is  constant  and  equal  to  the  sum  of  the 
squares  of  the  semi-axes. 

Proof  Let  (x',  «/')  be  the  co-ordinates  of  P  (last  figure), 
and  denote  the  semi-conjugate  axes  CP,  CD  by  a'  and  b' 
respectively.     Then  we  shall  have 

CP»  +  CD^  =  x'^  +  2/"  +  ^y--  +  l^;- 


_a'b'      aW 
~   b'   '^  a'  ' 


or 


a"  +  b"  =  a'-\-  b\ 


(9) 


12*7.  Problem.     To  find  the  angle  between  two  cofijugate 
axes. 

Let  6  and  6'  be  the  angles  which        ^ 
the  semi-conjugate  axes  make  with     '  "^ 
the  major  axis,  and  cp  the  angle  be- 
tween the  conjugate  axes.     Then 


and 


(p  =  e'  -e 


sin  cp  =  sin  6'  cos  0  —  sin  ^  cos  6 


Denote  the  semi-conjugate  axes  by  a'  and  b',  and  the  co- 
ordinates of  P  by  x',  y'.  Then  (§  125)  the  co-ordinates  of  D 
are 


-p'' 

+^'• 

Hence                 sin  Q  =  ^-; 

cos  u  =  — ,-; 

•     af       ^^' 

cos^'=       ,,- 

Substituting  in  (a),  we  have 

bx"    , 
^'"  ^  =  aa'b'  + 

THE  ELLIPSE. 


143 


(10) 


But  since  (?/',  y')  is  on  the  curve, 

Therefore  sm  a>  =     ,   ,,,  =  -nr- 

128.  Theorem  VII.  The  area  of  the  parallelogram  which 
touches  an  ellipse  at  the  ends  of  conjugate  diameters  is  constant 
and  equal  to  the  area  of  the  rectangle  ivhich  touches  the  elli2)se 
at  the  ends  of  the  axes. 


Proof  From  the  last  equation  we  have  a'b'  sin  (p  =  ab; 
but  a'b'  sin  cp  is  equal  to  the  area  of  the  parallelogram  CPQD, 
and  ab  is  equal  to  the  area  of  the  rectangle  CAEB;  therefore 
the  parallelogram  QRST  =  the  rectangle  EFGH,  which  is 
constant. 

Cor.  1.  The  triangle  CPD  is  equal  to  the  triangle  ACB, 
each  being  one  half  of  the  parallelograms  QC  and.  ^(7  respec- 
tively. 

Cor.  2.  If  P  denote  the  perpendicular  from  C  on  QT,  we 
have 

P  .  CD  =  area  of  CPQD 
=  ab. 

Therefore  P'  =  ^^  =  jr-r' 

But,  by  §  126,  b''  =  a'  +  b'  -  a'\ 

21,3 

Hence  P'  =f    ,   ,  "!; ti-  (H) 


144 


PLANE  ANALYTIC  OEOMETRT. 


129.  Problem.     To  find  the  equation  of  the  ellipse  re- 
ferred to  a  pair  of  conjugate  diameters  as  axes. 

Let  CP,  CD  be  any  two  conju- 
gate semi-diameters;  take  CP  for 
the  new  axis  of  X,  and  CD  for  the 
new  axis  of  Y',  let  the  angle 
ACP=a,  and  the  angle  A  CD  =13', 
{x,  y)  the  co-ordinates  of  any  point 
Q  of  the  ellipse  referred  to  rect- 
angular axes,  and  {x',  y')  the  co-ordinates  of  the  same  point 
referred  to  the  new  axes. 

The  formulae  for  passing  from  rectangular  to  oblique  axes 
are  (§  29) 

a;  =  a;'  cos  a:  +  y'  cos  /?; 
y  =  x'  sin  a  -\-  y'  sin  /?. 

But  since  {x,  y)  is  on  the  ellipse,  we  have 

a'y'  +  h'x'  =  a'b\ 
Eliminating  x  and  y  from  these  three  equations,  we  have, 
after  reduction, 

(a"  sin'a  +  h^  cosV)a;"  -f  (a«  sin'y5  +  h'  cos' /3)y'' 

+  2{a'  sin  asm/3  -\-  V  cos  a  cos  §)x'y'  =  c^W. 
But  since  CP  and  CD  are  conjugate  semi-diameters,  we  have, 
by  (8),  the  condition 


mm' 


or 


that  is, 


tan  a  tan  p  = 
sin  a  sin  /3 


a'' 
a'' 

.2  > 


cos  a  cos  ^  a' 

or  «'  sin  o'  sin  y^  +  ^^  cos  a  cos  ^  =  0, 

Therefore  the  coefficient  of  x'y'  vanishes  and  we  have 

(rt'  sin^a  +  Z>'cosV)a^"+  (a"  sin'/?  -f  J'  cos' /3)y''=  a'b\    (12) 
which  is  the  equation  of  the  ellipse  referred  to  the  new  axes. 
By  putting  y''  =  0,  we  get 

a'F 


a'  sin'a  -j-  b'  cos'a 


=  CP\ 


THE  ELLIPSE. 


145 


which  we  have  already  denoted  by  «".     In  a  smiihir  manner 
we  get 

which  we  have  denoted  by  b'"^. 
Hence  (12)  may  be  written 

—  4-^-1- 

or,  suppressing  the  accents  on  the  variables,  since  the  equation 
is  entirely  general. 


-1-  J/. 


(13) 


Comparing  this  with  (1),  we  see  that  the  equation  of  the 
curve  referred  to  the  major  and  minor  axes  is  only  a  particu- 
lar form  of  the  more  general  one  which  we  have  just  obtained. 
From  the  identity  of  form  in  (1)  and  (13)  we  see  that  the 
transformations  of  the  former  are  applicable  to  the  latter; 
therefore  it  follows  that  any  formulae  derived  from  the  equa- 
tion of  the  ellipse  by  processes  which  do  not  presuppose  the 
axes  to  be  rectangular  will  be  applicable  when  any  pair  of 
conjugate  semi-diameters  are  substituted  for  the  principal 
semi-axes. 

130.  Supplemental  Chords. 

Def.  The  two  straight  lines  drawn  from  any  point  on  an 
ellipse  to  the  extremities  of  any  diameter  are  called  supple- 
mental chords. 

If  the  diameter  is  the  major  axis,  the  chords  are  called 
principal  supplemental  chords. 

Relation  between  Two  Supplemen- 
tal Chords.  Let  PP'  be  any  diame- 
ter, andP§,  P'Q  two  supplemental 
chords;  {%',  y')  the  co-ordinates  of 
P,  and  therefore  (—  x',  —  y')  the 
co-ordinates  of  P',  and  {x,  y)  the. 
co-ordinates  of  Q.  Then  the  equa- 
tion of  the  line  PQ  may  be  written 

y  -  y'  ^m{x  -  x'), 


146 


PLANE  ANALYTIC  GEOMETRY. 


iind  the  equation  of  the  line  P' Q  may  be  written 

y  -\-  y'  =  fn\x  +  x')', 

whence,  by  multiplication, 

y'  -  y"^  mm'[x'  -  x"),  (a) 

But  since  the  points  {x,  y)  and  {x' ,  y')  are  on  the   curve, 
we  have 

a'y^  +  h'x'  =  a'b' 
and  ay  +  Fx"=z  a'b'; 

whence  a'(?/'  -  y'')  +  b^x'  -  x'')  =  0, 


or 


(i) 


Comparing  (a)  and  (b),  we  have 


b' 
mm'  = U-, 


which  is  the  condition  that  holds  for  conjugate  diameters 
whose  slopes  to  the  major  axis  are  7n  and  m'  respectively 
(§124);  therefore— 

Theorem  VIII.  If  any  chord  of  an  ellipse  is  parallel  to  a 
diameter,  the  supplemental  chord  is  parallel  to  the  conjugate 
difljneter. 

Relation  of  the  Ellipse  and  Circle. 

131.  Let  a  circle  be  described  on  the  major  axis  of  an 
ellipse  as  a  diameter;  its  equation 
referred  to  the  centre  as  origin  is 

yc'  =  a"  -  x\  (a) 

where   y^  represents   the  ordinate 
F'M, 

The  equation  of  the  ellipse  gives 


ye   =  ^(a^ 


x^),       (b) 


Comparing  (a)  and  (b),  we  have 

y?  =  ^^     "' 
b 


ye 


whence 


a 


TEE  ELLIPSE. 


147 


that  is,  the  ordinate  of  the  ellipse  at  any  point  is  found  by 
multiplying  the  ordinate  of  the  circle  by  the  constant  factor 

-.     Hence  we  have 
a 

Theorem  IX.  If  all  the  ordinates  of  a  circle  he  dimin- 
ished in  the  same  proportion^  the  circle  zuill  be  changed  into 
a?i  ellipse. 

133.  The  Eccentric  Angle. 

Def.  If  we  join  P  and  C,  the  centre  of  the  ellipse,  the 
angle  P'CA  is  called  the  eccentric  angle  of  the  point  P. 

Problem.  To  express  the  co-ordinates  of  any  point  of  the 
ellipse  in  terms  of  the  eccentric  angle  of  that  point. 

Let  the  eccentric  angle  =  9,  and  x,  y,  the  co-ordinates 
of  the  point  P.     Then,  since  P^G  =  AG,  we  shall  have 

X  =  a  cos  q)', 
y  =  -  P'M 


=  —  a  sin  (z?  =  §  sin  w. 

a  ^  ^ 

133.  Problem.  To  find  the  area  of  an  ellipse. 

Describe  a  circle  on  the  major 
axis  as  a  diameter,  which  we  can  con- 
ceive to  be  divided  into  any  num- 
ber of  equal  parts.  At  any  two  ad- 
jacent points,  as  M,  N,  draw  the 
common  ordinates  MP',  NQ',  and 
through  P  and  P'  draw  PH,  P'H' 
parallel  to  the  axis.  Let  the  ordinates 
PM,  P'M  be  denoted  by  y^,  and  y^ 
respectively.  Then,  since  the  rectangles  MH,  MR'  have  the 
same  breadth,  namely,  MN,  they  are  to  each  other  as  their 
heights  MP,  MP'-,  that  is, 


MH_ 
MH' 


(§131) 


In  the  same  way  it  may  be  shown  that  any  other  pair  of 
similar  rectangles  in  the  ellipse  and  circle   have  the  ratio  of 


148 


PLANE  ANALYTIC  GEOMETRY. 


1)  :  a,  and  therefore  the  sum  of  all  the  rectangles  in  the  ellipse 
is  to  the  sum  of  all  the  corresponding  rectangles  in  the  circle 
Sisb  :  a. 

Now  if  the  number  of  equal  parts  into  which  the  axis  is 
divided  be  increased  indefinitely,  the  sum  of  all  the  rectangles 
in  the  ellipse  will  approach  the  area  of  the  semi-ellipse  as  a 
limit,  and  the  sum  of  all  the  rectangles  in  the  circle  will  ap- 
proach the  area  of  the  semi-circle  as  a  limit. 

Therefore  we  shall  ultimately  have 

Area  of  the  ellipse   _   b 
Area  of  the  circle     ""  a 

But  the  area  of  the  circle  =  Tra';  therefore  we  shall  have 
the  area  of  the  ellipse  =  Ttab.     Hence: 

Theorem  X.  The  area  of  an  ellipse  is  a  mean  propor- 
tional between  the  areas  of  the  circles  described  on  the  major 
and  minor  axes. 


Tangents  and  Normals  to  an  Ellipse. 

134.  Problem.     To  find  the  equation  of  the  tangent  to 
an  ellipse  at  a  given  point. 

Let  x',  if  be  the  co-ordinates 
of  any  point  on  the  curve,  and 
x" ,  y"  the  co-ordinates  of  an 
adjacent  point  on  the  curve. 
The  equation  of  the  secant  pass- 
ing tlirough  the  points  x' ,  y' 
and  x",  y"  is,  by  §  45, 


V 


-,{x  -  X'), 


^  ^  X"    —   X' 

Since  {x',  y')  and  {x",  y")  are  on  the  ellipse,  we  have 
^^y"   -I-  ^^c"    =  a'b' 


0: 


(^0 


and 

a'y'^'  +  ^V"  =  ct'b'', 

therefore 

«'(/"  -  y'")  +  h\^""  -  ^'")  = 

whence 

y"  -  ?/'            b'     x."  +  x' 
-jo"  -  x'   "       «'  •  ll"  +  ?/' 

THE  ELLIPSE.  149 

Substituting  in  (a),  we  have,  for  the  equation  of  the  secant, 

y  -  y  - -^ -fr^i^  -  ^')-  i») 

Now  if  the  points  (a;',  y')  and  {x'\  ?/")  approach  each 
other  until  they  coincide,  the  secant  SS^  will  become  the  tan- 
gent TT\     We  shall  then  have  at  the  limit 

re"  =  x'         and         ^"  =  ?/'; 

hence  (b)  becomes 

/  ox.  -. 

y  -  y  =  -  -.-  ^(^  -  ^), 

a    y 

which  is  the  equation  of  the  tangent  at  the  point  x',  y'. 

This  equation  may  be  simplified  thus:  Multiply  by  a^y^ 
and  we  get 

a'yy'  +  h'xx'  =  a'y''  +  b'x'^ 

or  %J^yl.  =  \,  (15) 

a  0 

The  equation  of  the  tangent  may  also  be  expressed  inde- 
pendently of  the  coordinates  of  the  point  of  contact,  as  fol- 
lows: 

135.  Problem.     To  find  the  condition  that  the  line 
y  =  mx  -f-  h 
may  be  tangent  to  the  ellipse 

a'  ^  b' 
If  we  eliminate  y  between  these  equations,  we  have 
x^       {mx  -^  hy    _ 

or  (Z*'  +  a''m')x''  +  'Za'mhx  =  a\b''  -  h'),  {a) 

for  determining  the  abscissae  of  the  points  in  which  the  line 
intersects  the  ellipse.  Since  the  line  is  to  be  a  tangent  to  the 
ellipse,  the  two  values  of  the  abscissa  will  be  equal.    Now  the 


150 


PLANE  ANALYTIC  OEOMETRT. 


condition  that  this  equation  may  have  equal  roots  is,  by  the 
theory  of  quadratic  equations  (§  8), 

whence  h'  =  h'  +  ce^^\  (16) 

or  h    =  ±  Vb'  +  ci'm', 

the  required  condition. 

Substituting  this  value  of  h  m  the  given  equation  of  the 

line,  we  have  

y  =  mx  ±  Vb'  +  a'w'  (17) 

for  the  equation  of  the  tangent. 

Conversely,  every  equation  of  this  form  is  the  equation  of 
some  tangent  to  the  ellipse.  The  double  sign  shows  that  there 
will  always  be  two  tangents  having  a  given  slope. 

Eemark.  From  the  facility  with  which  this  equation  en- 
ables us  to  solve  many  problems  involving  the  use  of  the  equa- 
tion of  the  tangent,  it  is  sometimes  called  the  magical  equa- 
tion of  the  tangent. 

136.   The  8ubtangent. 

Def.  The  projection  on  the  axis  of  Xof  that  portion  of 
the  tangent  intercepted  between  the  point  of  contact  and  the 
axis  of  X  is  called  the  subtangent. 

To  find  where  the  tangent  intersects  the  axis  of  X,  we 
make  i/  =  0  in  the  equation  of 
the  tangent.      Thus  the  equa- 


tion  of  the  tangent  is 

~^N. 

a^'^  b'    -  ^'             1 

/iY\ 

Making  ?/  =  0,  we  have             I 

C 

'^  ii  j 

.  =  J=C..              ^ 

___^ 

Subtracting  CM  or  x\  we  have 

Subtangent  =  MT 
_a' 

J  _a'-x" 

Cor.    The  subtangent  is  independent    of   b',    hence  all 


THE  ELLIPSE.  161 

ellipses  described  on  a  common  major  axis  have  a  common 
suhtangent  for  any  given  abscissa  of  thepoi?its  of  coiitact. 


This  property  enables  us  to  draw  a  tangent  to  an  ellipse 
from  any  point  on  the  curve. 

Thus,  let  P  be  any  point  on  the  curve;  describe  a  circle  on 
^^' as  a  diameter,  and  produce  the  ordinate  PM  to  meet 
the  circle  in  Q.     Then  if  x'  is  the  abscissa  CM,  we  have 

Subtangent  of  ellipse  = 7 —  =  subtangent  of  circle  =  MT, 

X 

Hence  if  QThe  drawn  tangent  to  the  circle  and  meeting  A  A' 
produced  in  T,  then,  by  what  has  just  been  proved,  Twill  be 
the  foot  of  tangent  to  the  ellipse  at  P,  which  is  found  by  join- 
ing TP.  If  the  point  T  were  given,  we  would  first  draw 
TQ  tangent  to  the  circle,  and  from  the  point  of  contact  Q 
draw  the  ordinate  QM,  intersecting  the  ellipse  in  P,  the  re- 
quired point  of  contact;  and  by  joining  P  and  T  we  ^\ould 
have  the  required  tangent. 

137.   Tangent  through  a  Given  Point. 
Let  the  tangent  line  be  required  to  pass  through  a  given 
point  {x'j  y')]  we  shall  then  have  the  condition 

y'  =  mx'  +  7i,  (a) 

which,  combined  with  (16),  will  enable  us  to  determine  m  and 
h.     Equation  (a)  gives 

h'  =  y"  -  2mx'y'  +  m'x'\ 


152  PLANE  ANALYTIC  GEOMETRY. 

which,  substituted  in  (IG),  gives 

(«^  -  x")m'  4-  2x'y'm  -\- h'  -  y''  =  0; 


whence        m  =  '- ~ 7^ .  (18) 

Co  X 

Since  there  are  two  values  of  m,  two  tangents  to  an  ellijise 
can  be  drawn  through  a  given  point.  There  are  three  cases 
depending  on  the  position  of  the  point: 

L  If  the  position  of  the  point  is  such  that 

ay  +  i'x"  -  a^V  <  0, 

the  vakie  of  m  will  be  imaginary.     The  point  {x',  y')  will 
then  be  within  the  ellipse. 

II.  If  ahf  +  Vx''  -  aW  >  0,  the  two  values  will  be  real 
and  different. 

III.  If  ay  +  b'x''  -  a'b'  =  0,  the  point  (x/,  y')  will  be 
on  the  ellipse,  the  two  tangents  will  coincide,  and  the  equa- 
tion can  be  reduced  to  the  form  (1(3). 

138.  Problem.  To  find  the  locus  of  the  pomt  from 
loMcli  two  tangents  to  an  ellij^sc  mahe  a  right  angle  with  each 
other. 

Let  the  equations  of  the  tangents  be 

y  =  mx  -\-  VF  -\-  (i^nf;  (a) 

y  =  m'x  4-  Vb'  +  a'm'\  (b) 
Then  the  condition  to  be  fulfilled  is  (§  47) 

mm'  +  1  =  0.  (c) 

Eliminating  ??i'  from  (b)  and   (c),  the  equation   of  the   two 
tangents  will  be 

y  —  mx  =  Vb^  +  «^??i'; 
my  -\-  X     =  Va"^  -\-  ¥nf. 

Now,  what  we  want  is  the  locus  of  the  point  which  is  on 
both  tangents  at  once;  that  is,  the  locus  of  the  point  whose 
co-ordinates  satisfy  both  of  these  equations.  To  find  the  re- 
quired locus,  we  must  eliminate  m  from  the  equations,  which 
we  do  thus: 


THE  ELLIPSE  153 

Squaring  and  adding,  wc  have 

^uv  +  1).;^  +  (m=  +  l)f  =  {m'  +  l)(a'  +  F), 
or  x^  -{-  y"^  =  ft'  -f  b', 

which  is  the  equation  of  a  circle  whose  centre  is  at  the  origin 
and  whose  radius  is    Va^  -\-  h"^. 

We  thus  have  the  result:  If  we  slide  a  right  angh  around 
an  ellipse  so  that  its  sides  shall  continually  touch  the  ellipse,  its 
vertex  will  describe  a  circle  whose  radius  is  equal  to  the  dis- 
tance betiueen  the  ends  of  the  major  and  minor  axes. 

139.  Problem.  A  perpendicular  being  drawn  from 
either  focus  of  an  ellipse  upon  a  moving  tangent,  it  is  required 
to  find  the  locus  of  the  foot  of  the  perpendicular. 

Let 

y  =  mx  +  V¥-\-  d'm^  (a) 

be  the  equation  of  the  tangent.  The  equation  of  a  line  per- 
pendicular to  (a)  and  passing  through  the  focus  whose  co- 
ordinates are  ae  and  0  is 

^  -  -  ^^c-^  -  ^^^)-  (^) 

From  (a)  we  have 


y  —  7nx  =  VF  +  cc^m^^j 
and  from  (b),  my  -\-  x  =  ae. 

Squaring  and  adding,  we  get 

(x'  +  y')  (1  +  m')  =  b'-\-  ahn'  +  a'G" 
=  a'{l  +  m') 

(since  ^V  -(-  ^'  =  «'). 
Therefore  we  have 

^'  +  f  =  a\ 


the  equation  of  the  required  locus,  which  is  a  circle  described 
on  the  major  axis  of  the  ellipse.  The  same  result  is  obtained 
if  we  draw  the  perpendicular  from  the  other  focus. 

140.  Perpendiculars  from  the  Foci  upon  the  Tangent. 
Problem.     To  find  an  expression  for  the  length  of  the  per- 


154 


PLANE  ANALYTIC  GEOMETRY. 


pendicular  from  either  focus  upon  the  tangent  to  an  ellipse  at 
the  point  {x',  y'). 

Let  p  and  jt?'  be  the  perpen- 
diculars FQ,  F'R  respectively. 
The  equation  of  the  tangent  is 

Vx'x  +  a'y'y  -  a/V  =  0; 

and  since  the  co-ordinates  of  the 
foci  F  and  F^  are  {ae,  0)  and 
(—  ae,  0)  respectively,  we  shall 
have,  by  §4], 

ab^ex'  —  a^b^ 


^  =-:7i^ 


aF(ex'  —  a) 


and 


P' 


Vb'x''  +  ay        Vb'x"  -f  ay 
-  aVex'-  d'h'       -  ah\ex'  +  a) 


+  «y 


vv 


ay' 


(19) 


which  are  the  required  expressions  for  the  perpendiculars. 

Product  of  the  Perpendiculars  from  the  Foci  upon  the  sam 
Tangent,     We  find,  by  multiplication, 
^  _  a^i^i^ce  -  e'x'') 
PP    -    i^x"  +  a'y"  ~ 
_       a'h\a^  -  e'x'') 
"  l'x''-^a\a'b'-Jfx") 
^      a^b^a'  -  e'x'') 

Vx"  +  a\a^  -  o:'-") 
_      a\\-e''){a^-e'x") 
a\\-e'')x'''^a\a^-x''') 

[since  b^  =  a'{l  -  e')] 
__  a'(l  -  e'){a'  -  eV) 


=  i%  (20) 

an  expression  which  is  independent  of  the  co-ordinates  x^  and 

2/'- 

Hence: 

Theorem  XI.  The  rectangle  contained  by  the  perpen- 
diculars from  the  foci  upon  a  tangent  to  an  ellipse  is  con- 
stant and  eqtcal  to  the  square  of  the  semi-minor  axis. 


THE  ELLIPSE.  155 


For  tlie  ratio  of  tlie  perpendiculars  we  have 
p    _  ci  —  ea;' 
p'  ~~  a  -\-  ex' 

=  '-r  (§  130) 

Hence: 

Theorem:  XII.  Tlie  perpe7idiculars  from  the  foci  upon  tlie 
tangent  have  to  each  other  the  same  ratio  as  the  focal  radii  of 
the  point  oftangency. 

141.  The  Normal  to  an  Ellipse, 

Problem.  To  find  the  equation  of  the  normal  line  at  any 
point  of  an  ellipse. 

Let  x',  y'  be  the  co-ordinates  of  any  point  on  the  ellipse. 
Then,  by  §  134,  the  equation  of  the  tangent  at  that  point  is 


VV 


^  +  1^  =  1'  («) 

V-x'      ,    y 

or  y  = ~,x  H — ,, 

<^  y         y 
The  equation  of  a  line  through  x' ,  y'  and  perpendicuhxr  to 
{a)  is,  by  §  47, 

y-y'  =  %{.^-^%  (21) 

which  is  the  equation  of  the  normal  at  x\  y'. 

14z2.   The  Subnormal. 

Def.  That  portion  of  the  normal  line  intercepted  be- 
tween the  point  on  the  curve  and  the  axis  of  X  is  called  the 
normal,  and  its  projection  on  the  axis  of  X  is  called  the  sub- 
normal. 

To  find  where  the  normal  cuts  the  axis  of  X,  we  make 
i/  =  0  in  the  equation  of  the  normal;  then  we  get  (see  fig., 
§136) 

CX=x'{^-^\  =  e'x', 

Hence  the  subnormal 

iVi¥=  CM-  CN 

=  X'  —  .t'    1 -J    =  -r,x' 

\         a  J        a 
=  (1-  e')x\ 


156 


PLANE  ANALYTIC  OEOMETRY. 


143.  Theorem  XIII.     The  normal  at  any  point  on  a7i 
ellipse  bisects  the  angle   co7i- 

tained  by  the  focal  radii  of  ^ ^^^^P^-' 

that  point. 

Proof  Let  us  put  2p,  ip', 
the  angles  FFN  iind  F'FN 
respectively. 

By  the  theorem  of  sines, 
we  have 


sin  rp      _  F_N^ 
sill  FNF~  FF' 


sin  t/j'      _  F^N 
sin  FNF'  ~  'WF' 


{a) 


Now,  FF  and  F^P  are  the  focal  radii  whose  lengths  are 
given  by  the  equations  (4)  and  (5),  §120.  Also,  by  §§136 
and  142,  we  readily  find 

FN  =  ae  —  eV  =  e{a  —  ex')  =  er; 
F'N  =  ae  +  e^^•'  ==  e{a  -\-  ex')  —  er'\ 
whence  {a)  gives 

e  sin  FNF  =  sin  i",        e  sin  FNF'  —  sin  ^'; 
and  then,  since  sin  FNF'  =  sin  FNF,  we  have 

rp  =  r- 

Therefore  the  normal  P^  bisects  the  angle  FFF', 

Cor.  The  tangent  at  any  point  of  an  ellipse  bisects  the 
exte7'ior  angle  formed  by  the  focal  radii  of  that  point. 

For  if  one  of  the  focal  radii,  as  F'F,  be  produced  to  any 
point  g,  and  the  tangent  P T  be  drawn,  the  angles  F'FF, 
FFQ  are  supplementary;  and  since  NFT  is  a  right  angle  and 
PiY  bisects  the  angle  F'FF,  FT  also  bisects  the  angle  FFQ, 
which  is  the  exterior  angle  formed  by  the  focal  radii  FF,  F'F, 

Eemark.  If  a  ray  of  light  proceed  from  F  to  any  point 
F  on  the  ellipse,  it  will  be  reflected  to  F'.  For  this  reason 
the  points  P  and  F'  are  called /oa,  or  burning  points. 

The  theorem  just  proved  enables  us  to  draw  a  tangent  at 
any  point  on  an  ellipse.  Thus,  let  F  be  any  point  on  the 
curve;  draw  the  focal  radii  FF,  FF';  produce  one  of  them, 
as  FF',  and  bisect  the  exterior  angle  thus  formed  by  FT, 
which  is  the  tangent  required. 


THE  ELLIPSE.  157 

EXERCISES. 

OJUl 

1.  Show  that  there  is  a,  certciiii  segment  of  the  major  axis „v«>c..jl4 

of  an  ellipse  fronT^^mcn  normals  not  coincident  with  thatj^**^' 
axis  maybe  drawn  to  the  ellipse,  and  two  other  segments  from^-^ic^t-^ 
which  such  normals  cannot  be  drawn,  and  define  these  seg-^;;;;^ 
ments.  "^^ 

2.  Show  that  the  normals  from  three  or  more  equidistant 
points  on  the  major  axis  intersect  the  ellipse  in  points  whose 
abscissae  are  in  arithmetical  progression. 

3.  Show  that  the  ordinate  of  the  point  in  which  a  normal 

intersects  the  minor  axis  is  in  the  constant  ratio  ^ to 

e'  —  1 

that  of  the  point  where  it  intersects  the  ellipse. 

Reciprocal  Polar  Relations. 

144.   Chord  of  Contact. 

Def.  The  line  which  passes  through  the  points  where 
two  tangents  from  an  external  point  meet  an  ellipse  is  called 
the  chord  of  contact. 

Problem.     To  find  the  equation  of  the  chord  of  contact. 

Let  (7i,  h)  be  the  co-ordinates  of  the  point  from  which  the 
two  tangents  are  drawn;  (a:',  ?/'),  the  co-ordinates  of  the  point 
where  one  of  the  tangents  through  (7^,  h)  meets  the  curve, 
and  (a;",  y")  the  co-ordinates  of  the  point  where  the  other 
tangent  meets  the  curve. 

The  equation  of  the  tangent  at  (a:',  y')  is 

x'x       y'y  _ 

-^+-^-1,  (a) 

and  since  this  passes  through  (h,  k),  we  have 
hx'       hy'  _ 

Similarly,  __  _^  _^-  ^  1.  (c) 

Hence  it  follows  that  the  equation  of  the  chord  of  contact  is 
«=•  +  J.  -  i,  (^^) 


158 


PLANE  ANALYTIC  GEOMETRY. 


for  tliis  is  tlie  equation  of  a  straight  line,  and  is  satisfied  for 
X  —  x',  y  =  y'  and  x  =  x" ,  y  =  i/",  as  we  see  from  (b)  and 
(c). 

Cor.  From  what  has  been  shown  in  the  preceding  section, 
it  is  evident  tliat  this  equation,  referred  to  any  pair  of  conju- 
gate diameters  as  axes,  is 


hx       ky 


1. 


(23) 


145.  Locus  of  Interseciion  of  Two  Tangents.  het{x%y^) 
be  the  co-ordinates  of  any  fixed 
point  Q  through  which  the  chord 
of  contact  corresponding  to  the 
two  intersecting  tangents  is 
drawn;  (a;",  iy"),  the  co-ordinates 
of  P,  the  intersection  of  the  tan- 
gents. By  the  preceding  section, 
the  equation  of  the  chord  TT^  is 

x^       y^y_ 
a'   "^     b' 


1; 


but  since  (x\  y')  is  a  point  on  the  chord,  we  have  the  con- 
dition 

x^'x'       y"y'  _ 

a"    '^     b-"     ~  ^' 

which  the  co-ordinates  of  tlie  point  of  intersection  must  always 
satisfy.  Hence,  regarding  ic",  «/"  as  variables  and  omitting 
the  accents,  the  equation  of  the  locus  of  the  point  of  intersec- 
tion of  the  two  tangents  is 


(24) 


Cor,     This  equation,  referred  to  a  pair  of  conjugate  diam 
eters  as  axes,  will  be 

a'' 


-t-     ^,2 


(25) 


146.  Pole  and  Polar. 

The  identity  of  form  in  the  equations  of  the  iangenf,  the 
chord  of  contact  and  the  locus  of  the  intersection  of  tangents 


THE  ELLIPSE.  159 

drawn  from  the  extremities  of  chords  passing  through  a  fixed 
point  is  only  the  expression  of  a  reciprocal  relation  whicii 
exists  between  the  locus  and  the  fixed  point  (a?',  t/').  This 
relation  is  one  of  polar  reciprocity  and  is  expressed  by  the 
following  theorem: 

Theorem  XIV.  1.  If  chords  in  an  ellipse  he  draiun  through 
any  fixed  point  and  tangents  he  drawn  from  the  extremities  of 
each  chord,  the  locus  of  the  intersections  of  the  several  pairs 
of  tangents  will  he  a  straight  line. 

2.  Conversely,  If  from  different  points  in  a  straight  line 
pairs  of  tangents  he  drawn  to  an  ellipse,  their  chords  of  con- 
tact will  intersect  in  07ie  point. 

Defs.  The  straight  line  which  forms  the  locus  of  the 
intersection  of  two  tangents  drawn  from  the  extremities  of 
any  chord  which  passes  through  a  fixed  point  is  called  the 
polar  of  that  point. 

Reciprocally,  the  fixed  point  is  called  the  pole  of  the 
straight  line  which  forms  the  locus. 

Thus,  if  P  be  the  fixed  point  through  which  the  chords 
GG',  HH'  are  drawn,  and  pairs  of  tangents  GR,  G'R,  HQ, 
H' Q  be  drawn  from  their  extremities,  intersecting  in  R  and 
Q  respectively,  then  the  line  QR  is  the  polar  of  P,  and  P  is 
the  pole  of  QR.  If  the  polo  is 
on  the  curve  as  at  H,  then  the 
tangent  ^i?  is  the  polar;  and  if 
the  pole  is  without  the  curve,  as 
at  Q,  then  it  follows  that  the 
chord  of  contact  HH'  is  the, 
polar;  hence  we  see  that  the 
tayigent  and  the  chord  of  con- 
tact are  respectively  the  polars  of  the  point  of  contact  and  of 
the  intersection  of  the  tangents  drawn  from  the  extremities 
of  the  chord  of  contact. 

Hence  it  follows  that  if  {x',  y')  be  the  co-ordinates  of  any 
point  within,  on  or  without  the  curve,  the  equation  of  the 
polar  is 

$  +  ^^^  =  1,  (26) 


160  PLANE  ANALYTIC  GEOMETRY. 

or,  when  referred  to  a  pair  of  conjugate  diameters  as  axes, 

a''  +  f^n  -  1.  (^7) 

The  equation  of  the  diameter  conjugate  to  that  which 
passes  through  the  point  (:c',  y')  is 

^  +  ff  =  0, 

which  shows  that  the  diameter  and  the  polar  (27)  are  parallel; 
hence  we  have  the  following  theorem: 

The  polar  of  any  poi7it  in  respect  to  an  ellipse  is  parallel  to 
the  diameter  conj^igate  to  that  luhich  passes  through  the  point, 

147.  Polar s  of  Special  Points. 

Polar  of  the  Centre.  If  in  the  equation  of  the  polar  (26) 
we  suppose  the  pole  {x',  y')  to  approach  the  centre,  x'  and  y' 
will  approach  zero  as  their  limit,  and  one  or  both  the  co- 
ordinates, X  and  y,  of  any  point  of  the  polar  will  increase 
indefinitely.     Hence  tlie  polar  of  the  centre  is  at  infinity. 

This  is  also  seen  from  the  fact  that  tangents  at  the  ex- 
tremities of  any  diameter  meet  at  infinity. 

Polar  of  a  Point  on  o?ie  of  the  Axes.  When  y'  =  0,  we  get 

X  =   —,-=  'd  constant, 

which  shows  that  the  semi-major  axis  is  a  mean  proportional 
between  the  distances,  a;'  and  x,  of  the  pole  and  polar  from 
the  centre.  Since  the  same  reasoning  may  be  applied  to  a 
point  on  the  minor  axis,  we  conclude: 

Theorem  XV.  Either  semi-axis  is  a  mean  proportional 
ietiveen  the  distances  cut  off  from  it  hy  a  pole  upon  it,  and  ly 
the  corresponding  polar. 

Polar  of  the  Focus,  Substituting  for  {x',  y')  the  co-or- 
dinates of  either  focus  (±  ae,  0)  in  (26),  we  have 

x=  ±  -' 


TEE  ELLIPSE. 


161 


or,  the  polar  of  either  focus  of  an  ellipse  is  perpendimdar  to 
the  major  axis  and  at  a  distance  from  the  centre  equal  to  — 
measured  on  the  same  side  as  the  focus. 
148.  Directrix  of  an  Ellipse. 


If  DR  is  the  polar  of  the  focus  F,  we  have 
a 


and 


00  = 
DP 


OM 

00-  MO 

a 


_  a 
~  e 


ex 


but  from  the  linear  equation  of  the  curve  we  have 


hence 


DP  = 


FP 
FP 


FP 

and         -^-  = 


The  same  reasoning  applies  to  either  focus  and  its  polar. 
Hence: 

Theorem  XVI.  The  focal  distance  of  any  point  on  an 
ellipse  is  in  a  constant  ratio  to  its  distance  fj^o^n  the  polar  of 
the  corresponding  focus,  the  ratio  being  less  than  unity  and 
equal  to  the  eccentricity  of  the  curve. 

Def    The  polar  of  either  focus  is  called  a  directrix. 


162  PLANE  ANALYTie  OEOMETRT. 


EXERCISES. 

1.  Show  that  an  ellipse  has  a  pair  of  equal  conjugate 
diameters  whose  direction  coincides  with  the  diagonals  of 
the  rectangle  on  the  axes. 

2.  Show  that  the  equal  conjugate  diameters  of  an  ellipse 
bisect  the  lines  joining  the  extremities  of  the  axes. 

3.  Find  the  co-ordinates  of  the  point  in  an  ellipse  such 
that  the  tangent  there  is  equally  inclined  to  the  axes. 

a'  K' 

Ans. 


4.  If  r  and  r'  denote  the  focal  radii  of  any  point  on  an 
ellipse  whose  eccentric  angle  is  cp,  show  that 

r  =  «(1  —  e  cos  (p)        and        r'  =  a(i  -\-  e  cos  cp). 

5.  Find  the  equation  of  the  tangent  at  the  extremity  of 
the  latns  rectum.  A?is.  y  -\-  ex  =  a. 

6.  Find  the  equations  of  the  lines  joining  (1)  the  extremi- 
ties of  the  axes;  (2)  the  centre  and  the  extremities  of  the  latera 
recta. 

Ans.y=±-[x^a)-  .V  -  ± -.  •  -• 

7.  Find  the  equation  of  the  normal  at  the  extremity  of 
the  latus.  rectum. 

X 

Ans,  y  —  —  4-  ae'^  =  0. 

8.  If  the  normal  at  the  extremity  of  the  latus  rectum 
passes  through  the  extremity  of  the  minor  axis,  show  that  the 
eccentricity  of  the  ellipse  is  determined  by  the  equation 

e'-{-e'  -1  =  0. 

9.  Show  that  the  equation  of  the  tangent  at  any  point  is 

X  11 

—  cos  <z?  4-  4  sin  q>  —  1  =  0. 
a  b 

where  cp  is  the  eccentric  angle  of  tliat  point. 


THE  ELLIPSE.  163 

10.  Find  the  equation  of  the  straight  line  which  is  tan- 
gent to  the  ellipse  20?/^  +  bx"  =  100  at  the  point  (2,  2). 

11.  Tlirough  the  right-hand  focus  of  the  ellipse 
25y^  -{-  IGo;"  =  1600  is  drawn  a  focal  radius  making  an  angle 
of  30°  with  the  axis  of  X.  Find  the  equation  of  the  tangent 
to  the  ellipse  at  the  end  of  this  radius. 

12.  Express  the  intercepts  which  the  normal  to  an  ellipse 
cuts  off  from  the  co-ordinate  axes  in  terms  of  the  principal 
axes  of  the  ellipse  and  of  the  co-ordinates  of  the  point  (x^,  y^ 
in  which  the  normal  cuts  the  ellipse. 

13.  If  d  is  the  angle  which  a  radius  from  the  centre  of  the 
ellipse  forms  with  the  axis  of  X,  and  6'  the  angle  which  the 
tangent  to  the  ellipse  at  the  end  of  that  radius  forms  with  the 
same  axis,  find  what  relation  exists  between  6  and  6'.  ^»  ^  f>  ^^-'  P- 

14.  From  the  centre  of  an  ellipse  to  a  tangent  is  drawn  a 
line  parallel  to  the  focal  radius  of  the  point  of  tangency,  and 
meeting  the  tangent  at  the  point  p.  Find  the  locus  of  p  ^^ 
the  tangent  changes  its  position. 

15.  From  one  focus  of  an  ellipse  a  perpendicular  is  dropped 
upon  the  tangent  and  produced  to  an  equal  distance  on  the 
other  side.  Show  that  its  terminus  is  in  the  same  straight 
line  with  the  point  of  tangency  and  the  other  focus,  ^r  C^.  ^jf/A.^^ 

16.  The  same  thing  being  supposed,  find  the  locus  of  p 
when  the  tangent  moves  around  the  ellipse. 

17.  To  the  ellipse  a^y"^  +  5V  =  a^W  and  its  circumscribing 
circle  ?/'  -j-  2;'  =  c^  tangents  are  drawn  such  that  the  points  of 
tangency  shall  have  the  same  abscissa.  What  relation  exists 
between  the  subtangents,  and  what  relation  between  the  sub- 
normals? 

18.  Find  the  equations  of  the  tangents  drawn  from  the 
point  (0,  8)  to  the  ellipse  whose  equation  is  20?/^  -|-  hx^  =  100. 

19.  If  that  point  of  an  ellipse  to  which  a  normal  is  drawn 
approaches  indefinitely  near  to  the  major  axis,  what  limit  will 
the  intercept  of  the  normal  upon  the  axis  of  X approach? 

20.  On  the  major  axis  of  an  ellipse  a  point  is  taken  whose 
abscissa  is  a/.  Find  the  slope  and  equation  of  the  tangents 
from  this  point. 


164  PLANE  ANALYTIC  GEOMETRY. 

21.  At  what  points  will  the  tangents  which  make  an  angle 
of  45°  wich  the  principal  axes  cut  those  axes? 

22.  Find  the  intercept  upon  the  minor  axis  when  the 
normal  approaches  the  end  of  that  axis. 

23.  Find  the  equations  of  the  two  tangents  to  the  ellipse 
hy"^  -\-  Zx"  —  lb  which  are  parallel  to  the  line  3?/  —  4.r  -j-  1  =0. 

24.  To  the  ellipse  36?/'  +  25^:'  =  900  are  to  be  drawn  tan- 
gents cutting  the  axis  of  X  at  an  angle  of  30°.  Find  the  co- 
ordinates of  the  points  of  tangency. 

25.  Having  given  the  ellipse  J/x"  -\-  a^y"^  =  a'y'  and  the 
circle  x^  -\-  y"^  =i  ab,  it  is  required  to  find  the  equation  of  the 
common  tangent  to  the  two  curves.  Find  also  the  angle  at 
which  the  curves  intersect. 

26.  If  two  points  as  poles  be  taken  on  a  tangent  to  an 
ellipse,  where  will  their  polars  intersect? 

27.  The  chord  of  contact  to  two  tangents  of  an  ellipse  is 
required  to  pass  through  the  focus.  AVhat  is  the  locus  of  the 
point  where  the  tangents  intersect? 

28.  Find  the  pole  of  the  line  y  —  mx  +  li  with  respect  to 
the  ellipse  ay  +  Ux^  =  a^y^, 

29.  If  tangents  to  the  circumscribed  circle  of  an  ellipse  be 
taken  as  polars,  what  will  be  the  locus  of  the  pole? 

30.  Find  the  locus  of  the  pole  w^hen  the  polar  is  required 
to  be  a  tangent  to  the  circle  described  upon  the  minor  axis  of 
the  ellipse  as  a  diameter. 

31.  If  a  series  of  poles  be  taken  on  the  diameter  of  an 
ellipse,  show  that  the  polars  will  all  be  parallel  to  each  other. 

32.  If  chords  be  drawn  from  any  point  of  an  ellipse  to  the 
ends  of  either  principal  axis,  show  geometrically  that  they  are 
parallel  to  a  pair  of  conjugate  diameters. 

33.  If  a  line  of  fixed  length  slide  with  its  two  ends  con- 
stantly upon  the  respective  sides  of  a  right  angle,  show  that 
any  point  upon  it  describes  an  ellipse. 

34.  The  area  of  an  ellipse  is  to  be  equal  to  that  of  the  con- 
centric circle  passing  through  its  foci.     Find  its  eccentricity. 

.  )  1/5-1)4 


THE  ELLIPSE.  165 

35.  The  minor  caxis  of  an  ellipse  is  12,  and  its  area  is  equal 
to  that  of  a  circle  whose  diameter  is  20.     What  is  its  major 


axis 


36.  The  area  of  an  ellipse  is  equal  to  that  of  a  circle  cir- 
cumscribed around  the  square  upon  its  minor  axis.  Find  the 
angle  whose  sine  is  the  eccentricity.  Ans.  60°. 

37.  Show  that  the  equation  of  the  normal  at  the  point 
whose  eccentric  angle  is  cp  is 

ax  sec  cp  —  hy  cosec  cp  =  a^  —  If, 

38.  If  cp  and  q)'  be  the  eccentric  angles  of  any  two  points 
P ,  Q  on  an  ellipse,  sho\7  that  the  area  of  the  parallelogram 
formed    by  tangents  at    the  extremities   of    the   diameters 

through    P  and   Q  is  -^—, — -. r-     When  is   this  area  a 

°  ^       sin(<p'  —  cp) 

minimum? 

39.  Show  that  the  circle  described  on  any  focal  chord  as  a|  r 
diameter  touches  the  circle  described  on  the  major  axis  as  aj 
diameter.  _i- 

40.  Normals  are  drawn  to  an  ellipse  and  J;he  circumscrib- 
ing circle  at  points  having  the  same  abscissa.  Show  that  the 
locus  of  their  intersection  is  a  circle  whose  radius  is  a  -\-h. 

41.  Show  that  the  locus  of  the  intersection  of  tangents 
to  an  ellipse  at  the  extremities  of  conjugate  diameters  is  an 
ellipse. 

42.  Show  that  the  tangents  at  the  extremities  of  any  chord 
of  an  ellipse  meet  on  the  diameter  which  bisects  that  chord. 

43.  If  q)  and  q)'  denote  the  eccentric  angles  of  the  vertices 
of  two  conjugate  diameters  of  an  ellipse,  show  that 

tan  (p  tan  (p'  -\-l  =  0. 

44.  If  6  denote  the  angle  which  any  focal  chord  makes 
with  the  major  axis,  show  that  the  length  of  the  chord  is 

-TT. J r3T>  and  the  length  of  the  diameter  parallel  to  the 

w(  L         6    cos    t/ ) 

,      ,  .  2h 

chord  is  --J- ^ ^-^. 

i/(l  —  e'cos'^) 

45.  If  cp  and  cp'  be  the  eccentric  angles  of  any  two  points 


166  PLANE  ANALYTIC  GEOMETRY. 

on  an  ellipse,  show  that  the  equation  of  the  chord  which  joins 
the  points  is 

Z>cosi(«^+  cp').x-^a^mi{cp^  cp') .  y  —  abcosi((p  —  cp'). 

46.  Find  the  polar  equation  of  the  ellipse  (1)  when  the 
centre  is  the  pole,  and  (2)  when  the  left-hand  vertex  is  the 
pole,  the  major  axis  being  the  initial  line  in  both  cases. 

h'  %aV  cos  6 

A'/is.  r  = 5 5v.;         r  = 


e' COS'S'  a'sm'd-^d'cos'O 

47.  Show  that  the  perpendicular  from  the  centre  on  the 
chord  which  joins  the  extremities  of  two  perpendicular  diam- 
eters of  an  ellipse  is  of  constant  length. 

48.  Find  the  polar  co-ordinates  of  that  point  on  an  ellipse 
at  which  the  angle  between  the  radius  vector  and  tangent  is 
a  minimum.  A7is.  a;     cos~^e. 

49.  If  the  equation  x"^  -\-  y^  =  a^  represent  an  ellipse,  ex- 
press its  eccentricity  in  terms  of  the  angle  between  the  axes. 

50.  Show  that  the  sum  of  the  reciprocals  of  two  focal 
chords  at  right  angles  to  each  other  is  constant  and  equal  to 

g'  +  y 
2ab'   ' 

51.  A  tangent  is  inclined  to  the  major  axis  of  an  ellipse  at 
an  angle  6.  Show  that  the  rectangle  contained  by  perpendi- 
culars upon  it  from  the  ends  of  the  major  axis  varies  as  cos'0. 

52.  If  7\,  r,  1\  be  the  radii  vectores  corresponding  to  the 
angles  (9  -  60°,  ^,  ^  +  60°,  show  that 

111  1 


r,       r^       r       -J  latus  rectum* 

53.  Show  from  the  equation  ?/'  =  -J(2ax  —  x^)  and  from 

§  119  that  if  the  major  axis  of  an  ellipse  becomes  infinite 
while  the  parameter  remains  finite,  the  ellipse  will  become  a 
parabola. 

54.  Show  that  the  line  from  the  focus  to  the  point  of  in- 
tersection of  two  tangents  bisects  the  angle  formed  by  the 
focal  radii  of  the  points  of  tangency. 


CHAPTER  VII. 
THE    HYPERBOL 


Equation    and    Fundamental    Properties    of 
the    Hyperbola. 

149.  Def.  An  hyperbola  is  the  locus  of  a  point  the 
difference  of  whose  distances  from  two  fixed  points  is  con- 
stant. 

The  two  fixed  points  are  called  the  foci  of  the  hyper- 
bola. 

The  distances  from  any  point  on  the  curve  to  the  foci  are 
called  focal  radii,  oy  focal  distances. 

The  hyperbola  is  described  mechanically  as  follows:  Take  any  two 
fixed  points,  as  i^and  F',  and  at 
one  of  them,  as  F',  let  a  ruler  be 
pivoted,  while  to  the  other  point, 
F,  is  fastened  a  thread  whose 
length  is  less  tlian  that  of  the 
ruler. 

Attach  the  other  end  of  the 
thread  to  the  free  end  of  the  ruler 
at  J),  and  stretch  the  thread  tightly 
against  the  edge  of  the  ruler  with 
a  pencil-point,  P.  Then,  while  the  ruler  is  moved  round  the  pivot  at  F\ 
let  the  pencil-point  slide  along  the  edge  of  the  ruler  so  as  to  keep  the 
t-hread  lightly  stretched;  the  pencil-point  will  describe  an  hyperhola,  be- 
cause in  every  position  of  P  we  shall  have 

F'P  -  FP=  {F'P  +  PB)  -  (FP  -f  PD). 

But  F'P-\-  PD  is  the  length  of  the  ruler,  and  FP -\-  PD  is  the  length 
of  the  thread,  and  the  difference  between  the  lengths  of  these  is  con- 
stant; therefore  we  have 

F'P  -  FP=a.  constant, 
which  agrees  with  the  definition. 


168 


PLANE  ANALYTIC  GEOMETRY. 


By  interchanging  the  fixed  extremities  of  the  ruler  and  thread  we 
shall  obtain  a  second  figure  equal  and  similar  in  every  respect  to  the 
first,  but  turned  in  the  opposite  direction.  Thus  we  see  that  the  com- 
plete curve  consists  of  two  branches,  as  represented  above. 

150.  Problem.     To  find  the  equation  of  the  hyperlola. 

Let  tlie  straight  line  drawn 
through  the  foci"  ^e  taken 
as  the  axis  of  X;  t.  "*  point 
C  midivay  between  tj  ioci 
be  taken  as  the  origin,  and 
tlie  perpendicular  to  FF' 
through  C  as  the  axis  of  Y, 
Let  the  distance  between  the 
foci  =  2c;  the  difference  be- 
tween any  two  focal  radii  =  %a\  and  x,  y,  the  co-ordinates  of 
any  point  P.     Then  we  have 


and  therefore 


F'M  ^  x-^-c, 
FM  =x-c', 


(1) 


^^"  =  (^  +  ^r  +  !/'; 

PF^={x-cY^y^', 
and,  by  the  fundamental  property  of  the  curve, 

V(^  +  cf  4-  y'  -  f  (:,•  -  cy  +  y'  =  2a. 
Freeing  this  equation  of  surds,  we  have 

(c'  -  a')x'  -  a'y'  =  a\c'  -  a'), 
which  is  the  required  equation. 

This,  however,  may  be  simplified  by  putting,  for  the  sake 
of  brevity, 

c'-a'  =  b';  (2) 

hence  we  have 

h'x'  -  ay  =  a'b\  (3) 

or,  dividing  through  by  a'^»', 

which  is  the  equation  of  the  hyperbola  referred  to  its  centre 
and  axes. 


THE  HYPERBOLA. 


169 


151.  Relations  among  Axes  and  Foci  of  the  H}ipcr1)ola. 
If,  in  the  equation  (3),  we  put  y  =  0,   we  have,  for  the 
jioints  in  which  the  curve  cuts  the  axis  of  X, 

x=  ±a=  CA  or  CA'. 

Therefore  the  curve  cuts  the  axis  of  Xin  two  points,  A 
and  A',  equidistant  from  tlie 
origin  and  between  the  origin 
and  the  foci. 

Def.  The  points  A,  A' 
where  the  line  joining  the  foci 
cuts  the  curve  are  called  the 
vertices  of  the  hyperbola. 

The  line  ^^'  is  called  the 
transverse  axis. 

The  point  C  midway  between  the  vertices  is  called  the 
centre  of  the  curve. 

If  X  =  0,  we  have 


y  =  ±  b  V-  1, 

which  shows  that  the  curve  cuts  the  axis  of  l^in  two  imagi- 
nary points  situated  on  opposite  sides  of  the  centre  and  at 
the  imaginary  distance  b  V—  1  from  it. 

Measure  off  now  on  the  axis  of  l^the  distances  CB,  CB', 
each  equal  to  h,  the  real  factor  of  this  imaginary  value  of  ?/. 
Then: 

Def.  The  line  BB^  is  called  the  conjugate  axis  of  the 
hyperbola. 

Solving  the  equation  (3),  for  y,  gives 

y=±\  vi^^^\  (5) 


which  is  real  for  all  values  of  x  greater  than  a.  Hence,  when 
x'>  a,y  has  two  real  values  equal  in  magnitude  but  of  opposite 
signs;  therefore  the  curve  is  symmetrical  in  reference  to  the 
axis  of  X. 

If  X  <  a,  the  values  of  y  are  imaginary;  therefore  no  point 
of  the  curve  lies  nearer  to  the  centre  than  the  vertices. 


170  PLANE  ANALYTIC  GEOMETRY. 

If  X  increases  without  limit  in  either  direction,  y  increases 
without  limit,  and  therefore  the  curve  extends  indefinitely 
both  to  the  right  and  to  the  left  of  the  points  A,  A'. 

By  solving  the  equation  of  the  curve  for  x,  we  can  easily 
show  in  a  similar  manner  that  the  curve  is  symmetrical  in 
reference  to  the  axis  of  Y. 

Def.  The  distance  CF  =  CF'  =  c  of  each  focus  from 
the  centre  is  called  the  linear  eccentricity  of  the  hyper- 
bola. 

The  ratio  -  is  called  the  eccentricity  of  the  hyperbola, 
and  is  represented  by  the  symbol  e. 

Since                           c' =  a'' +  ^% 
we  have  e  = = — .  (6) 

Hence  the  eccentricity  of  an  hyperbola  is  always  greater 
than  unity. 

From  (6)  we  find 

l^a  V7~^\,  (7) 

which  expresses  the  semi-conjugate  axis  in  terms  of  the  semi- 
transverse  axis  and  the  eccentricity;  and  since  e  >  1,  l  may 
be  greater  or  less  than  a.  For  this  reason  we  do  not  use  the 
terms  major  and  7ninor  axis  as  in  the  case  of  the  ellipse. 

Cor.  By  comparing  the  equation  of  the  ellipse  with  that 
of  the  hyperbola,  we  see  that  the  equation  of  the  latter  may 
be  deduced  from  that  of  the  former  by  simply  writing  —  Z>' 
for  4-  'b\     Hence 

Any  function  of  b  i7i  the  ellipse  ivill  be  converted  into  the 
corresponding  function  in  the  hyperbola  by  merely  changing 
h  into  b  V  —  1. 

153.  Equilateral  Hyperbola.  An  hyperbola  in  which 
the  transverse  and  conjugate  axes  are  equal  is  called  an 
equilateral  hyperbola. 

From  (3)  we  see  that  the  equation  of  the  equilateral 
hyperbola  is 

x'  -n"^  a\ 


TUE  IIJTEIWOLA. 


171 


153.  Def.  The  parameter  or  lalus  rectum  of  an  hy- 
perbola is  the  chord  through  the  focus  perpendicular  to  the 
tiansverse  axis. 

Theorem  I.  The  parameter  of  an  hyperhola  is  a  third  pro- 
portional to  the  transverse  and  conjugate  axes. 

In  order  to  find  the  value  of  the  parameter  or  latus  rec- 
tum, we  put  a;  =  c  in  the  equation  of  the  curve.  The  equa- 
tion of  the  curve  may  be  written 

.'•=*' -4 

And  substituting  c  for  x  and  denoting  the  semi-parameter  by 
p,  we  have 


whence 


or 

that  is, 


ap  —  Jf', 
a  :  b  '.'.  b  \  p. 


(8) 


Cor.     The  length  of  the  semi-parameter,  in  terms  of  a  and 
e,  is 

p  =  a{e'  -  1). 

154.  Focal  Radii. 

Problem.     To  express  the  lengths  of  the  focal  radii  in 
terms  of  the  abscissa  of  the  point 
from  which  they  are  draiun. 

Let  r  and  r'  denote  the  focal 
radii  of  any  point  P  whose  co- 
oi-dinates  are  {x,  y).  Then,  from 
the  figure,  we  have 

(x  -  aey  +  f 


whence 


=  {x  -  aey  +  -^,[x'  -  a^) 

=  {x  -  aey  -{-  le'  ^l){x' 
=  e^x^  —  "Haex  +  «'; 
r  ~  ex  —  a.    ' 


«') 


(9) 


172 


PLANE  ANALYTIC  GEOMETRY, 


In  a  similar  manner  we  find 

r'  =  ex  +  a.  (10) 

Either  of  these  expressions,  being  of  one  dimension  in  a;, 
is  called  the  linear  equation  of  the  hyperbola. 

We  observe  that  their  difference  is  2a,  as  it  should  be. 

155.  Conjugate  Hyperhola. 

We  will  now  point  out  the  signification  of  the  line  BB', 
whose  length  is  2J  and  which  is  defined  as  the  conjugate  axis 
of  the  curve.  It  is  so  called  by  reason  of  the  important  rela- 
tion it  bears  to  a  companion-curve,  called  the  conjugate 
hyperbola,  whose  equation  we  will  now  develop. 

Let  an  hyperbola  be  described 
about  the  foci  G,  G'  situated  on 
the  axis  of  Y,  and  at  the  same 
distance  from  the  centre  (7 as  the 
foci  F,  F'  oi  the  hyperbola  which 
we  have  hitherto  been  consider- 
ing. Let  {Xj  y)  be  the  co-ordi- 
nates of  any  point  Q  on  this  new 
curve.  Then,  retaining  the  same 
origin  and  axes  of  reference  as  before,  we  shall  have 

CG  =  CG'  =  c,        x  =  NQ        and        y  =  CJST; 

therefore  G'Q'  =  (c  +  yY  +  q:'; 

GQ'  =  (c-yy^x\ 

Let  the  difference  between  the  focal  radii  G'Q,  GQ  be  2b 
instead  of  2a.  Then  we  shall  have,  by  the  definition  of  the 
curve 

^(c  +  yy  +  ^'  -  i^(o-yy-^x^  =  ^b, 

which,  when  freed  from  radicals,  becomes 

Z.V  -  (c'  -  b')y'  =  -  h\c'  -  b'). 
But  c'  -b'  =:^  a'; 

therefore  i'x'  -  ay  =  -  a'b% 


or 


^  _  .^L  -  _  1 
a'       b'  ~         ' 


(11) 


THE  HYPERBOLA.  178 

which  is  the  equation  of  the  companion -curve  or  conjugate 
hyperbola. 

If  in  (11)  we  put  x  =  0, 

y=±b=CB    or     CB', 

which  shows  that  the  conjugate  hyperbola  has  its  transverse 
axis  coinciding  in  direction  and  equal  in  magnitude  to  the 
conjugate  axis  of  the  primary  curve. 
If  ^  =  0,  we  have 


x=:  ±a  V—  1. 

But  CA  =  a  and  AA^  =  2a;  therefore  the  transverse  axis  of 
the  primary  curve  is  the  conjugate  axis  of  the  new  curve. 
Thus  we  see  that  what  is  called  the  co7ijugate  axis  of  an  hyper- 
bola is  in  fact  the  tj^ansverse  axis  of  the  conjugate  hyperbola. 

Def.  A  conjugate  hyperbola  is  one  which  has  the 
conjugate  axis  of  a  given  hyperbola  for  its  transverse  axis, 
and  the  transverse  axis  of  the  given  hyperbola  for  its  conju- 
gate axis. 

By  comparing  (4)  and  (11)  we  see  that  the  equations  of  an 
hyperbola  and  its  conjugate  differ  only  in  the  sign  of  the  con- 
stant term.  Since  the  conjugate  hyperbola  holds  the  same 
relation  to  the  axis  of  Fthat  the  original  does  to  the  axis  of  X, 
we  may  obtain  the  equation  of  the  former  from  that  of  the 
latter  by  simply  interchanging  the  quantities  which  relate  to 
the  two  axes.  Thus  if  the  equation  of  the  original  hyper- 
bola is 

b'x'  -  ay  =  a'b\ 

then,  by  interchanging 

X  and  y, 
a  and  b, 

we  have,  after  changing  signs, 

JV  -  ay  =  -  a'b', 
which  is  the  equation  of  the  conjugate  hyperbola. 


174 


PLANE  ANALYTIC  QEOMETRY. 


156.  Polar   Equation   of  the  Hyperhola,   the  left-hand 
focus  being  the  pole,  and  the  transverse  axis  the  initial  line. 


Let  the  angle  A'F'F  =  6  and  PF'  =  r  be  the  polar  co- 
ordinates of  any  point  P.     Then  we  shall  have 

PF'  =  PF"  +  FF''  -  2PF\  FF'  cos  FF'P, 
or         PF''  =  r=  +  4ft'e^  -  4aer  cos  6, 
Now,  by  the  fundamental  property  of  the  curve,  we  have 
PF-  PF'  ^'^a, 


or  ^v'  +  ^a'e' 

whence  we  get 

r 


4iaer  cos  6  —  r  =  2a; 
a(e'  -  1) 


(12) 


1  -i-  e  cos  8' 
which  is  the  equation  required. 

The  polar  equation  may  also  be  very  readily  obtained  from 
the  linear  equation  of  the  curve  in  the  same  manner  as  in  the 
case  of  the  ellipse.     (See  §  121,  Ellipse.) 

157.  To  trace  the  form  of  the  curve  from  its  polar  equa- 
tion. 

In  (12)  let  6  =  0.  Then  r  =  a{e-\)  =  F'A\  As  6 
increases  from  0  past  90°,  r  increases  and  becomes  infinite 
when  1  +  e  cos  6^  =  0 

1 


or  when 


cos  6 


Thus,  while  6  increases  from  0  to  the  angle  whose  cosine  is 


,  that  portion  of  the  curve  is  traced  out  which  begins  at 

c 


THE  UTPERBOLA.  175 

^'and  jiasses  through  P  to  an  indefinite  distance  from  the  ver- 
tex.   As  6  increases  from  the  angle  whose  cosine  is to  180°, 

G 

r  is  negative  and  decreases;  hence  the  portion  P' A  in  the 
lower  riglit-hand  quadrant  is  traced  out.  When  6  =  180°, 
r  —  —  rt(e+l)  =  F' A.    As  Q  increases  from  180°  to  the  angle 

whose  cosine  is in  the  third  quadrant,  r  is  negative  and 

increases  numerically,  and  becomes  indefinitely  great  when 

cos  ^  = .     Thus  the  portion  AP*'  is  traced  out.     As  d 

increases  from  that  angle  in  the  third  quadrant  whose  cosine 

is to  360°,  r  again  becomes  positive,  is  at  first  indefinitely 

e 

great  and  then  diminishes  until  d  =  360°,  when  r  =  a(e  —  1) 
=  F'A',  as  it  should.  Thus  the  portion  P'"A'  in  the  lower 
left-hand  quadrant  is  traced  out. 

EXERCISES. 

1.  Prove  the  following  propositions: 

I.  The  distance  of  each  focus  from  the  centre  is  ae. 
II.  The  distance  of  each  focus  from  the  nearer  vertex  is 
a{e  —  1),  and  from  the  farther  vertex  a{e  +  1). 

III.  The  distance  between  the  vertices  of  the  hyperbola 
and  of  its  conjugate  is  equal  to  that  between  the  centre  and 
the  foci. 

IV.  If  we  put  e'  for  the  eccentricity  of  the  conjugate 
hyperbola,  we  shall  have 

e"  +  e"  =  e''e'\ 

V.  The  eccentricity  of  an  equilateral  hyperbola  and  of  its 
conjugate  are  each  V2. 

2.  Find  the  semi-axes  and  eccentricity  of  the  hyperbola 

16a;'  -  V  =  144.  Ans.  a  =  3;     Z*  =  4;     e  =  |. 

o 

3.  Find  the  eccentricity  and  semi-parameter  of  the  hyper- 
bola  36a;'  -  25i/'  =  900.  Ans.  e  =  -^;    ^  =  7.2. 


176  PLANE  ANALYTIC  GEOMETRY. 

4.  What  is  the  equation  of  the  hyperbola  when  the  dis- 
tance between  the  foci  is  6  and  the  difference  of  the  focal 
radii  of  any  point  of  the  curve  is  4? 

Ans.  5x'  -  4if  =  20. 

5.  The  distance  from  the  focus  of  an  hyperbola  to  the  more 

remote  vertex  is  4  and  the  eccentricity  is  — .     Find  the  equa- 

o 

tion  of  the  curve  and  its  latus  rectum. 

Ans.  —x""  —  —li^  =  1;     latus  rectum  =  -— . 
9  4*^  3 

6.  What  is  the  equation  of  the  hyperbola  whose  transverse 
axis  is  10  and  whose  vertex  bisects  the  distance  between  the 
centre  and  the  focus?  Ans.  ox""  —  ?/'  =  75. 

7.  The  equation  of  an  hyberbola  is  x^  —  4:y^  =  12.  Find 
the  equation  of  the  conjugate  hyperbola  and  its  eccentricity. 

Ans.  x^  —  Ay^  =  —  12;     e  =  Vd. 

8.  If  e  and  e'  denote  the  eccentricity  of  an  hyperbola  and 
its  conjugate,  show  that 

e       h 


9.  Find  the  equation  of  the  hyperbola  when  the  left-hand 
focus  is  the  origin. 

Ans.  77, ^  H X  =  e  —  1. 

h        a        a 

10.  Show  that  by  multiplying  every  ordinate  y  of  an 
ellipse  referred  to  its  centre  and  axes  by  the  imaginary  unit 

V—  1,  it  will  be  changed  into  an  hyperbola  having  the  same 
axes. 

11.  A  line  parallel  to  the  transverse  axis  is  drawn  so  as  to 
intersect  both  an  hyperbola  and  its  conjugate.  Show  that  tlie 
segments  contained  between  the  two  hyperbolas  diminish 
indefinitely  as  the  line  recedes  indefinitely.  Also,  that  the 
rectangle  contained  by  one  of  those  segments,  and  by  the  sum 
of  the  two  segments,  one  of  which  is  cut  out  of  the  line  by 
each  hyperbola,  is  equal  to  the  square  upon  the  transverse 
axis. 


THE  HYPERBOLA. 


177 


Diameters  of  the  Hyperbola. 

158.  Def.  A  diameter  of  an  hyperbola  is  any  line 
passing  through  the  centre.  The  length  of  a  diameter  is 
the  distance  between  the  points  in  which  it  meets  the  curve. 

Theorem  II.  Every  diameter  of  an  hyperhola  or  of  its 
conjugate  is  bisected  hy  the  centre. 

Proof.  Let  the  equation  of  any  line  through  the  centre  of 
ail  hyperbola  be 

y  =  mx.  {a) 

The  equation  of  the  hyperbola  is 

b'x'  -  a'y'  =  a'b%  {b) 

and  the  equation  of  the  conjugate  hyperbola, 

b'x'  -  ay  =  -  a'b\  (c) 

Solving  (a)  and  (b)  for  x  and  y,  we  have 

ab 


and 


x=  ± 


y 


Vb'-a'm' 
mab 


(d) 


VU'-a'm'  ) 
And  solving  (a)  and  (c)  for  x  and  y,  we  have 

ab 


and 


ic  =  ± 


y 


Va'm'- 
mab 


(^) 


Va'7n'-  b^  J 

From  {d)  and  (e)  we  see  that  the  points  of  intersection  of 
the  line  y  =  mx  with  the  hyperbola  and  its  conjugate  are  at 
equal  distances  on  each  side  of  the  origin.     Q.  E.  D. 

.  When  F  >  a'^m'^    or    m  <  ±  -,  the  values  of  x  and  y  in  (d) 

are  real,  which  shows  that  the  line  (a)  intersects  the  given 
hyperbola  at  finite  distances  from  the  centre;  while  in  (e)  the 
values  of  x  and  y  are  imaginary,  Avhich  shows  that  the  line 
(a)  does  not  then  intersect  the  conjugate  hyperbola. 


178 


PLANE  ANALYTIC  GEOMETRY. 


If  J'  <  rf'w'    or    w  >  ±  -,  the  values  of  a;  and  y  in  (d) 

are  imaginary,  wliilc  in  {e)  they  are  rea?,  sliowing  that  the 
line  does  not  then  meet  the  given  hyperbola,  but  meets  the 
conjugate  at  finite  distances  from  the  centre. 

159.  Asymptotes.     If  J}"  —a^m^  or  m  =  ±  -,  the  values 

of  X  and  y  in  both  (4)  and  (5)  become  infinite.     Hence  the 

diameter  whose  slope  to  the  transverse  axis  is  either  A —  or 

a 

meets  the  hyi^erbola  or  its  conjugate  only  at  infinity. 

Def.  That  diameter  of  an  hyperbola  which  meets  the 
hyperbola  and  its  conjugate  at  infinity  is  called  an  asymp- 
tote of  the  hyperbola. 

Cor.  1.     The  equation  of  the  q  -^/^ 

asymptote  CP  is 

y  =  -X,       or   ^-1  =  0;  (13) 


and  of  the  asymptote  CQ, 
^=--„^,  or  f +  1  =  0.   (14) 


Cor.  2.  Equations    (13)   and 
(14)  are  the  equations  of  the  diagonals  of  the  rectangle  formed 
by  the  axes  of  the  curve.     Hence  : 

Theorem  III.  The  asymptotes  coi^icide  luith  the  diagonals 
of  the  rectaiigle  contained  hy  the  trayisverse  and  conjugate  axes. 

160.  Theorem  IV.  The  locus  of  the  centres  of  paral- 
lel chords  of  an  hyperhola  is  a  diameter. 

The  demonstration  of  this  theorem  is  similar  in  every  re- 
spect to  that  of  Theorem  III.  of  the  Ellipse.  Substituting 
—  V  for  y^  in  §123  of  the  Ellipse,  we  have,  omitting  the 
accents  on  the  variables, 

—  Ifx  -\-  c^my  —  0, 
or  7/  =  -^— .r,  (15) 


THE  HYPERBOLA.  179 

as  the  equation  of  the  locus  of  the  centres  of  parallel  chords. 
This  is  the  equation  of  a  straight  line  through  tlie  centre,  and 
is  therefore  a  diameter  of  the  curve.  By  giving  m  suitable 
values,  (15)  may  be  made  to  represent  any  line  through  the 
centre.     Hence 

Every  diameter  bisects  some  systein  of  parallel  chords. 

If  m'  be  the  slope  to  the  transverse  axis  of  any  diameter 
which  bisects  a  system  of  parallel  chords  whose  slope  is  7n, 
then  the  equation  of  the  diameter  is 


y 

=  m'x. 

But,  by  (15), 

y 

b' 
am 

is  also  the  equation 

of  the  diameter; 

therefore 

w' 

or 

mm' 

~  a'' 

(16) 

which  is  the  relation  which  must  hold  between  the  slope  of 
any  system  of  parallel  chords  and  the  slope  of  the  diameter 
which  bisects  these  chords.     Hence: 

Theorem  V.  If  one  diameter  bisects  chords  parallel  to  a 
second  diameter,  the  latter  ivill  bisect  all  chords  parallel  to  the 
former. 

161.   Conjugate  Diameters. 

Def.     Two  diameters  are  said  to  be  conjugate  to  each 
other  when  each  bisects  all  the  chords  parallel  to  the  other. 
The  equation  of   condition   for   conjugate  diameters   is, 

by  (16), 

mm'  =  -^, 
a 

where  m  and  m'  denote  their  respective  slopes  to  the  trans- 
verse axis.  Since  the  second  member  of  this  equation  is  posi- 
tive, m  and  m'  must  have  the  same  signs;  that  is,  they  must 
be  both  positive  or  both  negative.  Hence  the  angles  which 
conjugate  diameters  make  with  the  transverse  axis  must  be 
both  acAite  or  both  obtuse. 


180 


PLANE  ANALYTIC  GEOMETRY. 


If 


and  if 


m  < 

a' 

m'> 

a' 

m> 

b 

a' 

m'< 

b 
a' 

m  = 

-|. 

m'  = 

a 

Whence  it  follows  that  the  conjugate  diameters  of  an  hyperbola 
lie  on  the  same  side  of  the  conj^igate  axis,  but  on  opposite 
sides  of  an  asymptote;  so  that  if  one  of  two  conjugates,  as  PP\ 


meets  the  hyperbola,  the  other,  QQ'y  will  meet  the  conjugate 
hyperbola.  PP'  produced  bisects  all  chords  parallel  to  QQ' 
in  either  branch  of  the  hyperbola,  and  QQ\  produced  if  neces- 
sary, bisects  all  chords  drawn  between  the  two  branches  of  the 
curve  and  parallel  to  PP'. 

Conversely,  QQ'  produced  bisects  all  chords  of  either 
branch  of  the  conjugate  hyperbola  parallel  to  PP',  and  PP' 
produced  bisects  all  chords  parallel  to  QQ'  behveen  the  two 
branches  of  the  conjugate  hyperbola. 

Cor,  Since  the  chords  of  a  set  become  indefinitely  short 
near  the  extremity  of  the  bisecting  diameter,  they  will  coin- 
cide in  direction  with  the  tangent  at  that  point.     Hence: 

Theorem  VI.  The  tangent  to  an  hyperbola  at  the  end  of  a 
diameter  is  parallel  to  the  conjugate  diameter. 

162.  Pkoblem.  Give7i  the  co-ordinates  of  the  extremity 
of  one  diameter,  to  find  those  of  either  extremity  of  the  conju- 
gate diameter. 


THE  HYPERBOLA. 


181 


Let  PP'  and  QQ'  be  any  two 
conjugate  diameters,  and  {x\  y') 
the  co-ordinates  of  P. 

The  equation  of  CP  is 

y  =  S^^  since  w  =  ^„  (a) 
and  the  equation  of  QQ'  is 


y  =  -r-^ 


or 


2/ 


and  the  equation  of  the  conjugate  hyperbola  is 

h'x'  -  ay  =  -  a'b\ 
Solving  (h)  and  (c),  we  have,  since  b^x'^  —  a^y^^  =  aW, 


(J) 


(«) 


and 


ft 


163.  Theorem  VII.  The  difference  of  the  squares  of  two 
conjugate  semi-dia^neters  is  constant  and  equal  to  the  differ- 
ence of  the  sqicares  of  the  semi-axes. 

Proof  Let  {x',  y')  be  the  co-ordinates  of  P  (last  figure). 
Then  the  co-ordinates  of  Q  will  be 

If  the  semi-conjugates  be  denoted  by  a'  and  h',  we  have 

CP^  -  CQ^  =  (z"  +  /^)  -{~y''  +  5-^") 


h'x' 


ay       b'x''  -  a'y' 


or 


b' 
l'-^  =  0"  -  b' 
—  a  constant. 


(17) 


182  PLANE  ANALYTIC  OEOMETRT. 

164.  Problem.  To  express  the  angle  hetween  two  conju- 
gate diameters  in  terms  of  their  lengths. 

Let  6  and  d'  denote  the  angles  which  the  conjugate  semi- 
diameters  CP,  CQ  make  with  the  transverse  axis,  and  (p  the 
angle  PCQ  between  them. 

Then  (p=e'  -e 

and  sin  cp  =  sin  0'  cos  6  —  sin  6  cos  &'.  (a) 

Now  if  (x',  y')  be  the  co-ordinates  of  P,  those  of  Q  will  be 


(jy'-/)- 


therefore 

sin  6  = 

cos  d 

x' 
~  a'' 

sin  (9'  = 

ix' 
'  aV 

cos  6' 

ay' 
~  bb'' 

Substituting 

in  (a),  we 

get 

sin  (p  = 

I'x'^  -  aSf' 
aha'b'      ~ 

ab 
■  a'b" 

the  required 
Cor,     Fr 

expression, 
om  (17)  we 

!  have 

a  constant; 

(18) 


therefore  a'  and  b'  increase  together  or  decrease  together. 
Hence,  when  each  tends  to  coincide  with  the  asymptote,  the 
product  a'b'  tends  towards  infinity,  and  sin  cp  tends  towards 
0;  therefore  the  angle  between  two  conjugates  diminishes 
Avithout  limit.  When  the  conjugates  coincide  with  the  asymp- 
totes, each  becomes  infinite. 

165.  Theorem  VIII.  T7ie  area  of  the  parallelogram 
whose  sides  touch  an  hyperbola  at  the  ends  of  any  pair  of  con- 
jugate diameters  is  consta7it  and  eqiial  to  the  rectatigle  formed 
by  the  axes  of  the  curve. 

Proof   From  (18)  we  have 

4:a'b'  sin  (p  =  4^5 

=  a  constant,  (19) 

which  proves  the  proposition. 


THE  HYPERBOLA. 


183 


166.  Problem.     To  find  the  equation  of  the  hyperhola 
referred  to  a2)air  ofcovjvgate 
diameters  as  axes. 

Let  DD',  HH'  be  any  pair 
of  conjugate  diameters.  Take 
DD'  for  the  new  axis  of  X, 
and  HH'  for  the  new  axis  of 
Yy  and  let  the  angle  XCD=  a 
and  XCH  =  /?.  We  may 
now  transform 

b'x'  -  ay  =  a'b' 

from  rectangular  to  oblique  axes  by  the  process  of  §  129,  or 
we  may  simply  change  b'  into  —  b'  in  equation  (12)  of  that 
section.     Thus  we  get 


{a'  shi'a  -  b'  cos'a)x''-{-  {a'  shi'/3-  b'  cos' /3)y" 


a'b\  (20) 


which  is  the  equation  required. 

By  putting  x'  and  y'  each  equal  to  zero,  we  get  the  inter- 
cepts on  the  axes  or  the  lengths  of  the  semi-conjugates. 
Thus,  when  y"  =  0, 


x'' 

= 

— 

a'b' 

= 

CD'  =  a"; 

a' 

sii] 

i'a 

—  b'  cos 

'a 

and 

when 

x" 

= 

0, 

we 

have 

/' 

— 



aW 

i  ~ 

-  CH'  = 

^,2c 

ir.2 

R         h^nn 

^2/; 

h". 


(a) 


(i) 


Because  the  new  axis  of  Xmeets  the  given  hyperbola,  the  now 

axis  of  1^  will  not  meet  the  curve,  but  will  meet  the  conjugate 

—  a'b' 
hyperbola.     Therefore  ^g^.^.,^ ^^^^^^^  is  a  negative  quan- 


a'sm'fi-b'cos'/3 


titv. 


From  (a)  and  (b)  we  get 


a'  sin' a  —  b'  cos' a  =  — 


a'b' 


and 


a'sm'P-b'cos'P 


a^ 
~b"' 


184 


PLANE  ANALYTIC  GEOMETRY. 


Substituting   in   (20)   and    dividing   by    —    d'b"^,    we    have, 
omitting  the  accents  from  the  variables, 

Also,  the  equation  of  the  conjugate  hyperbola  referred  to 
tlie  same  axes  is 


X'   _  f 


Tangent  and  Normal  to  an  Hyperbola. 

167.  Problem.  To  find  the  equation  of  the  tangent  to 
an  hyperbola. 

In  order  to  obtain  the  equation  of  the  tangent,  we  have 
only  to  repeat  the  process  of  §§  134,  135,  changing  b"^  into 


b'\     Thus  we  get 


or 


xx' 


b'x'x  =  -  a'b% 

Ml    -1 
b'      ~    ' 


(22) 


and  also  y  =  mx  ±  y  a'm^  —  ^>%  (23) 

where  m  is  the  slope  of  the  tangent  to  the  transverse  axis. 

Intercept  of  the  Ta7igent 
on  the  Axis  of  X. 

In   (22)    make    y  =   0. 
Then 


^=~  =  CT, 


(24) 


from  which  we  see  that  x 

and  x'  must  always  have  the 

same  sign;  and  since  x  is  always  positive  in  the  right  branch 

of  the  curve,  the  tangent  to  that  branch  always  intersects  the 

axis  of  X  to  the  right  of  the  centre. 

168.  Subtangent.     Foj-  the  length  of  the  subtangent  we 
liave,  from  the  figure, 

Subtangent  =  MT 


(25) 


THE  HYPERBOLA.  185 

169.  Theorem  IX.  The  ta?igent  to  an  hyperbola  at  any 
point  bisects  the  angle  formed  by  the  focal  radii  of  that  point, 

a^ 
Proof  Since  F'C=FC=ae  and  CT  =  ~,  we  hiive 


*  •  "V  •   ■ 

X' 

rT=ae-^^,  =  ^-Xex'^a) 

and 

FT=cte-^^,  =  ^-,{ex'-ay, 

whence 

F'T      ex'-\-a 
FT  ~  ex-a 

_F'P 

pp.  (§154) 

Therefore,  since  the  base  of  the  triangle  F'FP  is  divided  pro- 
portionally to  its  sides  (Geom.),  the  tangent  P2^  bisects  the 
angle  FPF', 

170.  Tangent  through  a  Given  Point. 

Let  (/i,  ^•)  be  the  co-ordinates  of  the  given  point,  and  {x' ,y') 
the  co-ordinates  of  the  point  of  contact.  The  equation  of  the 
tangent  is 

b'^x'x  —  a^y'y  =  ^'Z*'; 

but  since  the  tangent  must  pass  through  {li,  k)  and  {x\  y'), 
we  have 

bVix'  -  a'ky'  =  a'b%  (a) 

and  also  b'x''  -  a'y"   =  a''b\  (b) 

Eliminating  y'  from  these  equations,  we  have 

{a'k'  -  bVL')x"  +  2a'bVix'  -  a*{b'  +  F)  =  0; 
whence 

,  _  aV/h  Ta'k  V¥k'  -  b'¥  +  a'b'  ,^^. 

"" ¥¥^ln^ •  ^^^^ 

Since  x'  has  two  values,  two  tangents  to  an  hyperbola  can 
be  drawn  through  a  given  fixed  point.  The  tangents  will  be 
real,  coincident  or  imaginary  according  as 

a'F -Z»7i'  +  a'<^'>,     =     or     <     0; 
that  is,  according  as  the  given  point  is  luithin,  on  or  outside 
the  curve. 


186  PLANE  ANALYTIC  GEOMETRY. 

171.  Problem.  To  find  the  criterion  that  the  ttoo  tan- 
gents from  a  given  point  shall  touch  the  same  branch  of  the 
hyperbola. 

If  the  tangents  belong  to  the  5<ime  branch  of  the  curve,  the 
abscissae  of  the  points  of  contact  x'  will  have  like  signs;  but 
if  they  belong  to  opposite  branches,  unlike  signs.  Now,  in 
order  that  the  values  of  x'  in  (26)  may  have  like  signs,  we 
must  have  numerically 


a'bVi  >  a'k  Va'k'  -  b'h'  +  a'b'-, 
whence,  by  reduction. 


k  <  -h, 
a 


But  y  =  —X 

*^        a 

is  the  equation  of  the  asymptote;  and  if  we  take  on  the 
asymptote  a  point  whose  abscissa  x  is  equal  to  the  abscissa  It 
of  the  point  from  which  two  tangents  may  be  drawn,  we  shall 
have  k  y  y\  that  is,  the  ordinate  of  the  point  from  which 
two  tangents  can  be  drawn  to  the  same  branch  of  an  hyperbola 
must  be  less  than  the  corresponding  ordinate  of  the  asymptote. 
Hence  the  point  from  which  two  tangents  can  be  drawn  to  the 
same  branch  of  an  hyperbola  must  lie  in  the  space  between  the 
asymptotes  and  the  adjacent  branch  of  the  curve,  which  is  the 
required  criterion. 

Hence,  also,  if  the  point  lie  without  this  space,  the  two 
tangents  will  touch  different  branches  of  the  curve. 

173.  Problem.  To  find  the  locus  of  the  point  from 
which  two  tangents  to  an  hyperbola  make  a  right  angle  with 
each  other. 

The  solution  is  similar  to  the  corresponding  problem  in 
the  Ellipse.  We  will  therefore  simply  change  b"^  to  —  b"^  in 
the  process  of  §  138,  and  we  get 

x"  -\-y'  =  a'  -  b' 
for  the  required  locus,  which   is  a  circle  having  the  same 
centre  as  that  of  the  hyperbola  and  whose  radius  is  Va'^  —  b'\ 


THE  HYPERBOLA.  187 

Cor.  Two  tangents  at  riglit  angles  to  each  other  cannot 
be  drawn  to  an  hyperbohi  when  b  >  a. 

173.  Problem.  To  find  the  locus  of  the  intersection  of 
the  tangent  with  the  jjcrpendicular  on  it  from  the  focus. 

The  sohition  is  the  same  as  that  of  the  corresponding 
problem  in  the  case  of  the  ellipse. 

The  equation  of  tlie  required  locus  is  found  to  be 

^''  +  ^'  =  «% 
which  is  a  circle  described  on  the  transverse  axis  as  a  diameter. 

174.  Problem.  To  find  the  length  of  the  peiyendiciilar 
from  either  focus  upon  the  tangent  to  an  hyperbola. 

If  {x',  y')  be  the  co-ordinates  of  the  point  of  t^ingency, 
and  p,  p'  the  perpendiculars  from  the  foci  F  and  F'  respec- 
tively, we  find,  in  the  same  manner  as  in  the  Ellipse, 

—    ^(^'i^^'  ~  ^) 

,  _    ab'jex'-i-a)  ,   j  ^   '^ 

^   ~   Vb'x''  4-  ay'  J 
whence  we  get,  by  reduction, 

pp'  =  b'  (28) 

,^d  ^,-  =  ?^  =:  ^,  (29) 

2?         ex  -\-  a      r 

where  r  and  r'  denote  the  focal  radii  of  the  point  of  contact. 
From  the  last  two  equations  we  readily  find 

y  =  %b\        p''  =  -b' 

and  f  =  ^-.  (30) 

175.  Normal  to  an  Hyperbola. 

Problem.  To  find  the  equation  of  the  normal  to  an  hyper- 
bola. 

The  equation  of  the  normal  is  found  by  changing  b"  into 
—  h'  in  the  process  of   §  141.     Thus,  if  (x',  y')  be  the  co- 


188 


PLANE  ANALYTIC  GEOMETRY. 


ordiiiates  of  any  j^oint  P  on  the  hyperbola,  the  equation  of 
the  normal  PN  is 


y-y'  =  -i'i[''-'')' 


(31) 


The  Subnormal. 


X  = 


Putting  y 


0  in  (31),  we  find 
CN-  CM=  {e'-l)x\ 


x'  =  e'x'  = 


Hence,  subnormal  =  3IN 

176.  Theorem  X.  The  normal  at  a7iy  point  of  an  hyper- 
bola bisects  the  external  angle  contained  by  the  focal  radii  of 
that  point. 

Since  the  angles  FPF'  and  FP H  ^yq  supplementary  and 
TP  bisects  FPF',  therefore  PN,  which  is  perpendicular  to 
TP,  must  bisect  the  external  angle  FPH. 

Cor.  Comparing  this  result  with  that  of  §  143,  we  see  that 
if  an  ellipse  and  an  hyperbola  have  the  same  foci,  the  curves 
loill  intersect  at  right  angles. 

For  at  the  point  of  intersection  the  tangent  of  one  will  be 
the  normal  of  the  other,  and  vice  versa. 

Remark.  The  student  should  note  the  relations  between  the  differ- 
ent theorems  and  formulae  relating  to  the  ellipse  and  the  corresponding 
ones  relating  to  the  hyperbola.  Where  the  formula  of  the  one  class  con- 
tains the  symbol  Ij^,  it  may  be  applied  immediately  to  the  other  by  chang- 
ing the  sign  of  5^,  which  will  be  the  result  of  substituting  b  V —\  for  b. 
Where  only  the  first  power  of  b  enters, the  theorems  of  one  class  involving 
real  quantities  will  be  imaginary  when  transferred  to  the  other  class.  Thus 
w^e  have  imaginary  asymptotes  to  the  ellipse.  Tlie  apparent  exceptions 
arise  from  our  substituting  a  real  for  an  imaginary  conjugate  axis  in  the 
hyperbola  and  thus  referring  several  expressions  which  would  have  been 
imaginary  to  the  conjugate  hyperbola,  which,  it  must  be  remembered, 
is  not  a  part  of  the  curve  at  all. 


THE  HYPERBOLA.  189 


Poles  and  Polars. 

177.  Pkoblem.  To  find  the  equation  of  the  chord  oj 
contact  of  two  tangents  from  the  savie  given  point. 

Let  (7i,  k)  be  the  co-ordinates  of  the  fixed  point  from 
which  the  two  tangents  that  determine  the  chord  are  drawn. 
Then,  by  simply  changing  the  sign  of  Z>'  in  §  144,  the  equa- 
tion of  the  hyperbolic  chord  of  contact  will  be 


h'x      k'y  _ 
when  referred  to  the  axes  of  the  curve. 


(32) 


'$-^^1  =  ^  (33) 

when  referred  to  any  pair  of  conjugate  diameters. 

178.  Locus  of  Lnter section  of  Two  Tangeyits  whose  chord 
of  contact  passes  throicgh  a  fixed  point. 

Let  {x' ,  y')  be  the  co-ordinates  of  any  fixed  point  through 
which  the  chord  of  contact  belonging  to  any  two  intersecting 
tangents  is  drawn.  Then,  by  simply  changing  the  sign  of  y 
in  the  process  of  §  145,  we  shall  have,  for  the  equation  of  the 
required  locus, 

•'^-^-l-  (34) 

or,  when  referred  to  a  pair  of  conjugate  diameters, 

x'x      y'y  _  .^.. 

which  is  the  equation  of  a  straight  line,  the  polar  of  the 
point  {x',  y'). 

Cor.  The  student  may  easily  show,  as  in  the  case  of  the 
ellipse,  that  the  polar  of  any  point  in  respect  to  an  hyperbola 
is  parallel  to  the  diameter  conjugate  to  that  which  passes 
through  the  point. 


190 


PLANE  ANALYTIC  GEOMETRY. 


1*79.  Polar s  of  Special  Points. 

Polar  of  the  Centre,  Proceeding  in  the  same  manner  as 
in  the  ellipse,  we  find  that  the  polar  of  the  centre  is  at  in- 
finity. 

Tlie  Polar  of  any  Point  on  a  Diameter  A  is  a  straight  line 
parallel  to  the  conjugate  diameter,  and  cutting  the  diameter 
A  at  a  distance  from  the  centre  equal  to  the  square  of  the 
semi-diameter  on  which  the  point  is  taken  divided  by  the 
distance  of  the  point  from  the  centre. 

Polar  of  the  Focus.  Substituting  {±ae,  0)  for  {x/  y')  in 
the  equation  of  the  polar,  we  have 

X  =  ±-. 
e 

Hence  the  polar  of  the  focus  of  ayi  hyperlola  is  the  perpe^i- 

dicular  which  cuts  the  transverse  axis  at  a  distance  —  from 

e  -' 

the  centre  on  the  same  side  as  the  focus. 

180.  Distance  of  any  Point  on  the  Curve  from  either  Fo- 
cal Polar. 

Let  DRhe  the  polar  of  the  focus  F.     Then  we  have 

00='^; 

e 

DP  =  OM 

=  CM-  CO 


ex 


FP 


FP 

therefore  y^p  =  e. 

Whence: 

Theorem  XI.  The  focal  distance  of  any  point  on  an  hyper- 
bola is  in  a  coristant  ratio  to  its  distance  from  the  polar  of  the 
focus. 


THE  nYPERBOLA. 


191 


This  ratio  is  greater  than  unity  and  equal  to  the  eccentri- 
city of  the  curve. 

Def.  The  polar  of  the  focus  is  called  the  directrix  of 
the  hyperbola. 

The  above  property  enables  us  to  describe  the  curve  by  continuous 
motion,  as  follows:  Take  any  fixed 
straight  line  NR  and  any  fixed  point  F, 
and  against  the  former  fasten  a  ruler, 
and  place  another  ruler,  right-angled  at 
N,  so  that  its  edge,  NH,  may  move  freely 
along  NR.  At  F  attach  one  end  of  a 
thread  equal  in  length  to  the  hypothe- 
nuse  HQ  of  the  ruler,  and  the  other  end 
to  the  extremity  Q  of  the  ruler.  Then 
with  a  pencil-point  P  stretch  the  thread 
tightly  against  the  edge  HQ,  while  the 
ruler  is  moved  along  the  other  ruler, iViJ. 
The  point  P  will  describe  an  hyperbola, 
for  in  every  position  we  shall  have 


and  therefore 


=  a  constant. 


PF=  PR; 

PF  _  PR 
PI)'~  PD 

_m 

NQ 

181.  Cor.  From  §§  97,  148  and  180  it  follows  that  we 
may  define  a  conic  section  as  the  locus  of  a  point  which  moves 
in  such  a  way  that  its  distance  from  a  fixed  point  (the  focus) 
is  in  a  constant  ratio  to  its  distance  from  a  fixed  straight 
line  (the  directrix). 


PF 

In  the  ellipse,        the  ratio  -p-^ 

PF 
In  the  parabola,     the  ratio  -p^ 

PF 

In  the  hyperbola,  the  ratio  -pj: 


<  1. 


=  1. 


>  1. 


In  all  cases  this  ratio  is  the  eccentricity  of  the  curve. 

In  the  case  of  the  ellipse  and  hyperbola  there  is  a  direc- 
trix corresponding  to  each  focus.  In  the  case  of  the  parabola 
the  second  focus  and  directrix  are  at  infinity. 


192  PLANE  ANALYTIC  GEOMETRY. 

The  Asymptotes. 

183.  We  have  already  shown  (§159)  thit  the  equations 
of  the  asymptotes  when  re- 
ferred to  rectangular  co-ordi-     ^ 
nates  are 

a       b 

Now  since  the  equation  of 
the  hyperbola  referred  to  a 
pair  of  conjugate  diameters 
as  axes  is  of  the  same  form  as    -^^  ^  \^H 

when  referred  to  rectangular  axes,  we  at  once  infer  that  equa- 
tions {a)  transformed  to  the  same  conjugate  diameters  become 

a'   "^  b'  ~  ^' 

that  is,  the  equations  of  the  asymptotes  MG,  LH  when  re- 
ferred to  any  pair  of  conjugate  diameters  are  respectively 

and  i^+t'  =  °-  (") 

Equation  {h)  is  the  equation  of  a  line  which  passes  through 
the  centre  or  origin  and  the  point  {-{-  a',  -\-  h')',  that  is, 
through  C  and  2);  and  (c)  is  the  equation  of  a  line  which 
passes  through  the  origin  and  the  point  (-f  a',  —  h')  or  C 
and  E.     Hence  we  conclude: 

Theorem  XI.  The  asymptotes  comcide  in  direction  with 
the  diagonals  of  the  parallelograin  formed  ly  any  pair  of 
conjugate  diatneters. 

183.  Angle  between  the  Asymptotes. 

Let  GCX=a.     Then  tan  «  =  -; 

a 

b 
whence  sin  a  =  — ,  and        cos  a 


X 

— 

y 
b' 

X 

+ 

y 

Va'  +  b'  Va'  +  b' 


THE  HYPERBOLA.  193 

Now  since  XCH  =  GCX,  sin  GGH=.  sin  %GCX', 
hence  sin  GGH  =  2  sin  G^CXcos  GGX 


a'  +  b'' 

Cor.     In  the  equilateral  hyperbola,  a  =  h; 

hence  sin  GCH  —  1; 

that  is,  the  asymptotes  of  the  equilateral  hyperbola  intersect 
at  right  angles.  For  this  reason  the  equilateral  hyperbola  is 
sometimes  called  the  rectangular  hyperbola. 

184.  Theorem  XII.  Tlie  asymptotes  of  the  hyperMa 
are  its  tangents  at  infinity. 

We  prove  this  by  showing  that,  as  the  point  of  tangency 
on  an  hyperbola  recedes  indefinitely,  the  tangent  approaches 
the  asymptote  as  its  limit. 

1.  If,  in  equation  (24)  of  §  167, 

_a' 

we  suppose  x'  to  increase  without  limit;  x,  the  abscissa  of  the 
point  in  which  the  tangent  intersects  the  transyerse  axis,  ap- 
proaches zero  as  its  limit.  Hence  the  tangent  at  infinity 
passes  through  the  centre  of  the  hjrperbola. 

2.  From  the  equation  of  the  tangent, 

h'^x'x  —  a'y'y  —  a^V, 

Vx' 
it  follows  that  its  slope  to  the  axis  of  X  is  -^-,.     We  must 

asy 

now  find  the  value  of  this  slope  when  the  point  of  tangency 
{x* ,  y')  recedes  to  infinity.  Because  this  point  remains  on 
the  hyperbola,  we  have 

x^  _  ir  _ 
a'        b^  ~  ^' 


whence 


r: =/«+#)■ 


194  PLANE  ANALYTIC  GEOMETRY. 

As  y'  recedes  to  infinity,— 5  approaches  zero  as  its  limit;  whence, 

3^'  (I 

at  infinity,--^  =  ±  -p  and  we  have 

Slope  of  tangency  at  infinity  =  a:  -. 

Hence  the  tangents  at  infinity  are  a  pair  of  lines  whose  equa- 
tions are 

y=±-x, 

which  lines  are  the  asymptotes,  by  definition. 

185.  Problem.  To  find  the  equation  of  the  hyperlola 
referred  to  its  asymptotes  as  axes. 

Let  the  asymptote  CH  be  the  new  axis  of  X,  and  the 
other,  CG,  the  new  axis  of  Y\  {x,  y)  be  the  co-ordinates  of  any 
point  on  the  curve  referred  to  the  old  axes,  and  (x',  y')  the 
co-ordinates  of  the  same  point  referred  to  the  new  axes. 

The  equation  of  the  curve  referred  to  the  old  axes  is 

h'x'  -  aY  =  cc'b\  (a) 

which  must  be  transformed  to  the  new  or  oblique  axes,  the 
origin  remaining  the  same. 

The  formulae  of  transformation  are 

X  =  x'  cos  a  -{-  y'  cos  /?; )  /^x 

y  =  x'  sin  a  -\-  y'  sin  /?;  f 

where  a  and  /?  are  the  angles  which  the  new  axes  make  with 
the  old  axis  of  X\  that  is,  a  =  XCH  and  ^  =  GCX, 
or  /3  =  —  a. 

Therefore  (b)  becomes 

X  =  (x'  -\-  ?/')cos  a; 

y  =  {x'  —  ?/')sin  «; 

which  being  substituted  in  {a)  give,  after  obvious  reductions, 

(^'cos'o:-  a"  sin'a)(a;"+?/")+  ^b''  cos'a'+  a^  sin' a)x'y'=a''b\ 

-D   ,  ^  b  sin  a        b 

But  tan  a  =  —,       or =  -: 

a  cos  a        a 

whence  b^  cosV  =  «'  sin'<a'; 


THE  HYPERBOLA.  195 

which  being  substituted  in  the  preceding  equation  gives,  after 
dividing  by  a^ 

4  sin'or  x'y'  =  Z>'. 
But  sin  a  =  -  .    ^        ;  (§  183) 

therefore  4a;'?/'  =  a"*  +  Z>% 

or,  omitting  the  accents  on  the  variables,  since  the  equation 
is  perfectly  general, 

^y  =  '^.  (36) 

which  is  the  required  equation. 

Cor.  The  equation  of  the  conjugate  hyperbola  referred 
to  the  same  axes  is  readily  found  to  be 

-^  =  -^'.  m 

186.  Problem.  To  find  the  equation  of  the  tangent  to 
an  hyjwrhola  referred  to  the  asymptotes  as  axes. 

Let  {x',  y')  and  (re",  «/")  be  the  co-ordinates  of  any  two 
points  on  the  curve.  The  equation  of  the  secant  through 
these  two  points  is 

y  -y'  =  l^  1 1^  -  ^').  («) 

Since  the  points  (x^,  y')  and  {x",  y")  are  on  the  curve, 

a'  +  l>" 


^y  = 


and  X  y 


.,.,.  _«'  +  *' 


//.i' 


whence         x'y'  =  x"y",       or        y"  =  —j-,- 
which  substituted  in  (a)  gives,  after  reduction, 

y  -y'  =  -  ^(^  -  ^')- 


196 


PLANE  ANALYTIC  GEOMETRY. 


Now  at  the  limit,  x"  =  x'  and  the  secant  becomes  a  tan- 
gent; hence  the  equation  of  the  tangent  at  the  point  {x\  y')  is 


y 


whence 


or 


-y'--=- 

x'y  -\-  xy' 

=  3^'y', 

x'  ^  y' 

-2, 

(38) 


which  is  the  simplest  form  of  the  required  equation. 

Co7\  1.  Making  x  and  y  successively  equal  to  0,  we  get  the 
intercepts  on  the  axes; 

thus,    x=:2x'  =  CT 

and      y  =  2y'=  CT, 

Hence  the  point  of  contact 
is  the  middle  point  of  TT'\ 
or,  that  portion  of  a  tan- 
gent intercepted  between 
the  asymptotes  is  bisected 
at  the  point  of  contact. 

Cor.  2.  CT  X  Cr  =  4:x'y'=  a'  +  h'; 

or,  the  rectangle  formed  by  the  intercepts  cut  off  by  any  tan- 
gent from  the  asymptotes  is  constant  and  equal  to  the  sum 
of  the  squares  of  the  semi-axes. 

Cor.  3.  The  area  of  the  triangle  CTT'  is 

=  iCT.Cr  .  sin  Tcr 

^        a"  -{-  b" 
a'  +  i'         2ab 


2        •  a^  +  b' 
=  ab,  a  constant ; 

or,  the  area  of  the  triangle  formed  by  any  tangent  and  the 
asymptotes  is  constant  and  equal  to  the  rectangle  of  the 
semi-axes. 


THE  HYPERBOLA.  197 


EXERCISES. 

1.  Find  the  equation  of  that  hyperbola  whose  transverse 
axis  is  8  and  which  passes  through  the  point  (10,  25). 

Ans.  zrx  —       ■'     —  ^ 


16       2500 

2.  What  condition  must  the  eccentricity  of  an  hyperbola 
fulfil  in  order  that  the  abscissa  of  some  point  upon  it  shall  be 
equal  to  the  ordinate?  A?is.  e  <  V2. 

3.  Express  the  distance  from  the  centre  of  an  hyperbola  to 
the  end  of  its  parameter  in  terms  of  the  semi-transverse  axis 
and  eccentricity. 

4.  Show  that  each  ordinate  of  an  equilateral  hyperbola 
is  a  mean  proportional  between  the  sum  and  difference  of  the 
abscissa  and  semi-transverse  axis. 

5.  Write  the  equation  of  a  focal  chord  cutting  an  hyperbola 
at  the  point  (ic',  y'). 

6.  Find  that  point  upon  the  conjugate  axis  from  which  the 
two  tangents  to  an  hyperbola  form  a  right  angle  with  each 
other. 

7.  Where  do  the  tangents  drawn  from  a  vertex  of  the  con- 
jugate hyperbola  touch  the  hyperbola,  and  what  are  the  equa- 
tions of  these  tangents?  Show  that  they  are  bisected  by  the 
transverse  axis. 

8.  What  must  be  the  eccentricity  in  order  that  the  tan- 
gent at  the  end  of  the  parameter  may  pass  through  the  vertex 
of  the  conjugate  hyperbola? 

9.  Find  those  tangents  to  an  hyperbola  which  make  an 
angle  of  60°  with  the  transverse  axis. 

10.  What  must  be  the  eccentricity  of  an  hyperbola  that  the 
subnormal  may  always  be  equal  to  the  abscissa  of  the  point 
from  which  the  normal  is  drawn? 

11.  Find  the  equation  of  the  hyjierbola  when  the  origin  is 
transferred  to  one  of  the  vertices,  while  the  axes  of  co- 
ordinates remain  parallel  to  the  principal  axes. 

12.  Express  the  product  of  the  segments  into  which  a 


198  PLANE  ANALYTIC  GEOMETRY. 

focal  chord  is  divided  by  the  focus  in  terms  of  the  angle 
which  the  chord  forms  with  the  major  axis. 

Ans. 


1  -  e'^cos''^  • 

13.  Show  that  the  sum  of  the  reciprocals  of  the  two  seg- 
ments of  a  focal  chord  is  equal  to  four  times  the  reciprocal  of 
the  parameter. 

14.  The  line  x  =  dy  \s  n  diameter  of  the  hyperbola 
25x'  —  16y'  =  400.  Find  the  equation  of  the  conjugate 
diameter. 

15.  For  what  point  of  an  hyperbola  are  the  subtangent  and 
subnormal  equal  to  each  other? 

16.  Express  the  length  of  the  tangent  at  the  point  {x',  y'). 

17.  Find  the  condition  that  the  line  -^  -f  ^  =  1  shall  touch 
the  hyperbola  -^  —  -^  =  1.  Ans.     e"  —  e^  —  1. 

18.  A  perpendicular  is  drawn  from  the  focus  of  an  hyper- 
bola to  an  asymptote.  Show  that  its  foot  is  at  distances  a 
and  h  from  the  centre  and  focus  respectively. 

19.  Show  that  the  linear  equation  of  the  right-hand  branch 
of  the  hyperbola  when  a  focus  is  the  origin  is 

r  =:  ea;  =F  «(1  —  e')- 

20.  Each  ordinate  of  an  hyperbola  is  produced  until  it  is 
equal  to  the  focal  radius  of  the  point  to  which  it  belongs. 
Find  the  locus  of  its  extremity. 

21.  Find  the  equation  of  the  tangent  at  the  extremity  of 
the  latus  rectum. 

22.  Show  that  the  intercepts  cut  off  from  the  normal  by 
the  axes  are  in  the  ratio  of  ct^  :  5^ 

23.  In  an  hyperbola,  3fl  =  2c.     Find  the  eccentricity  and 

the  angle  between  the  asymptotes. 

A  3        .    _,4    .- 

Ans.     e  =  — ;    sm    ^  —  r  5. 
z  y 

24.  Show  that  the  angle  between  the  asymptotes  of  an 

hyperbola  is 

2  sec~^  e. 


THE  HYPERBOLA.  199 

25.  From  any  point  on  an  hyperbola  perpendiculars  are 
drawn  to  the  asymptotes.    Show  that  their  product  is  constant 

and  equal  to  -^— - — t^. 
^  a  -{-  b 

26.  From  any  point  in  one  of  the  branches  of  the  conju- 
gate hyperbola  tangents  are  drawn  to  an  hyperbola.  Show 
that  the  chord  of  contact  touches  the  other  branch  of  the 
conjugate  hyperbola. 

27.  Show  that  the  polar  equations  of  the  right-hand  branch 
of  an  hyperbola  referred  to  the  foci  are 

r  —  —^ ^        and        r  =  —^ ^. 

1  —  e  cos  a  1  —  e  cos  6 

28.  Show  that  the  polar  equation  of  the  hyperbola  when 
the  centre  is  the  pole  is 

,^ ^ 

^        e'  cos"  6-1' 

29.  Show  that  the  length  of  any  focal  chord  of  an  hyper- 

2  y^ 

bola  is  -  .  -5 ^-7^ -,  where  6  is    the  inclination  of  the 

a      e  cos  6—1 

chord  to  the  transverse  axis. 

30.  In  the  figure  of  §  182,  show  that  the  diagonal  P(>  is 
parallel  to  the  asymptote. 

31.  In  an  equilateral  hyperbola,  if  cp  is  the  inclination  of 
a  diameter  passing  through  any  point  P,  and  ^'  the  inclina- 
tion of  the  polar  of  P,  show  that 

tan  cp  tan  q)'  =  1. 

32.  Through  the  point  (5,  3)  is  to  be  drawn  a  chord  to  the 
hyperbola  25a:''  —  IQy^  =z  400  which  shall  be  bisected  by  the 
point.     Find  the  equation  of  the  chord. 

33.  In  an  hyperbola  is  to  be  inscribed  (or  escribed)  an  equi- 
lateral triangle,  one  of  whose  vertices  shall  be  at  the  right- 
hand  vertex  of  the  curve.  Find  the  sides  of  the  triangle, 
and  find  the  eccentricity  when  they  are  infinite. 

34.  Express  the  tangent  of  the  angle  between  the  two 
focal  radii  drawn  to  the  point  (^',  «/')  of  an  hyperbola,  and 


200  PLANE  ANALYTIC  GEOMETRY. 

thence  find  those  points  of  the  curve  from  which  these  radii 
subtend  a  right  angle. 

Ans.,  m  part,  tan  q>  =  -j^—, — y^ r-r. 

^  ^       a:"  -[-  2/     —  « e 

35.  For  what  point  on  an  equilateral  hyperbola  is  the  pro- 
duct of  the  focal  radii  equal  to  db"^? 

36.  From  the  foot  of  any  ordinate  of  an  hyperbola  a  tangent 
is  drawn  to  that  circle  described  upon  the  major  axis  as  a 
diameter.  Show  that  the  ratio  of  the  ordinate  to  the  tangent 
is  a  constant  and  equal  to  e"^  —  1. 

37.  Find  the  lengths  which  the  directrix  of  an  hyperbola 
cuts  off  from  the  asymptotes,  and  the  length  of  that  segment 
of  the  directrix  contained  between  the  asymptotes. 

Ans.   a  and  i  ~-  e. 

38.  Find  the  polar  of  the  vertex  of  the  conjugate  hyper- 
bola relatively  to  the  principal  hyperbola. 

39.  From  any  point  of  an  hyperbola  is  drawn  a  parallel  to 
the  asymptote,  terminating  at  the  directrix.  Find  the  ratio 
of  the  length  of  this  parallel  to  the  focal  radius  of  the  point, 
and  show  that  it  is  a  constant. 

40.  Show  (1)  that  the  sides  of  the  quadrilateral  whose  ver- 
tices are  at  the  termini  of  any  pair  of  conjugate  diameters  are 
equally  inclined  to  the  principal  axes;  (2)  that  all  such  quad- 
rilaterals in  the  same  hyperbola  have  their  corresponding  sides 
parallel  and  are  equal  in  area. 

41.  Find  that  point  of  an  hyperbola  for  which  the  tangent 
is  double  the  normal. 

42.  At  what  angle  does  the  hyperbola  x"  —  y""  =  a"  inter- 
sect the  circle  x^  -\-  y^  =  Oa'^? 

43.  A  line  drawn  perpendicular  to  the   transverse  axis  of 

an  hyperbola  meets  the  curve  and  its  conjugate  in  P  and  Q 

respectively.    Find  the  loci  of  the  intersection  of  the  normals, 

and  of  the  tangents,  at  P  and  Q. 

11^       x^         Vx^ 
Ans.     The  transverse  axis;  ~  —  ~  =  4--^^. 

b         a^         a^y 

44.  The  two  sides  of  a  constant  angle  slide  along  a  para- 
bola. Find  the  locus  of  the  vertex  of  the  angle,  and  compare 
the  cases  of  two  loci  whose  angles  are  supplementary. 


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CHAPTER    VIII. 

THE  GENERAL  EQUATION  OF  THE    SECOND    DEGREE. 

187.  Tiie  most  general  equation  of  the  second  degree  be- 
tween two  variables  x  and  y  may  be  written  in  the  form 

7W.7;'  +  ny"  +  %lxy  +  %])x  +  ^y  +  fZ  =  0;  (1) 

the  six  coefficients  m,  n,  I,  p,  q  and  d  being  any  constants 
w^iatever.  * 

We  may  divide  the  equation  throughout  by  any  one  of  the 
coefficients  without  changing  the  relation  between  x  and  y. 
One  of  the  six  coefficients  will  then  be  reduced  to  unity. 
Hence  the  six  coefficients  are  really  equivalent  to  but  five  in- 
dependent quantities. 

The  problem  now^  before  us  is:  What  possible  curves  may 
be  the  locus  of  the  general  equation,  and  what  common  pro- 
perties have  these  curves? 

One  property  may  be  recognized  at  once  by  determining 
the  points  of  intersection  of  the  curve  with  a  straight  line. 
Let  the  equation  of  the  straight  line  be 

y  z=zhx-\-  h. 

By  substituting  this  value  of  y  in  (1)  we  shall  have  an 
equation  of  the  second  degree  in  x  whose  roots  will  give  the 
abscissas  of  the  points  of  intersection.  Now,  since  an  equa- 
tion of  the  second  degree  always  has  two  roots  which  may  be 
real,  equal  or  imaginary,  we  conclude: 


*  Three  of  these  terms  are  "written  with  the  coefficient  2  because 
many  expressions  which  enter  into  the  theory^  especially  when  deter- 
minants are  introduced,  are  thus  simplified.  We  then  consider  I,  p  and 
q  as  representing  one  half  the  coeflacients  of  xi/,  x  and  y  respectively. 


202  PLANE  ANALYTIC  OEOMETRY. 

Theorem  I.  Every  straight  Ime  intersects  a  curve  of  the 
second  degree  in  two  real,  coincident,  or  imaginary  poi^its, 

188.  Change  of  Origin,  To  continue  the  investigation, 
we  change  the  origin  of  co-ordinates  without  changing  the 
form  of  the  curve.  If  we  put  x'  and  y'  for  the  co-ordinates 
referred  to  the  new  origin,  we  must,  in  equation  (1),  put 

X  =  x'  -\-  a', 

y  =  y'  +  V; 

a'  and  h'  being  the  co-ordinates  of  the  new  origin,  which  are 
to  be  determined  in  such  a  way  as  to  simplify  the  equation. 
Making  this  substitution,  the  equation  becomes 

mx'''-\-  2lx'y'  +  ny'''  +  2(a'm  +  h'l  +  p)x'^  2(a'l  +  b'n  +  q)y' 
+  a''7n  +  b''n  +  2a'b'l  +  2a'p  -f  'Z^q  +  c?  =  0.         (2) 

We  now  so  determine  the  co-ordinates  a'  and  b'  that  the 
coefficients  of  x^  and  y'  shall  vanish.     To  effect  this  we  have 

the  equations 

nm'  +  lb'  =-p',)  (^) 

la'  +  nb'  =  -q;)  ^  ^ 

in  which  «'  and  b^  are  the  unknown  quantities.     Solving  the 

equations,  we  find 

.  _np-lq^^ 

^        72  >      I 

I  -mn   [  .  (3) 

r  —  mn  J 

Omitting  for  the  present  the  special  case  in  which  ?  —  mn 
=  0,  these  values  of  a'  and  b'  will  always  be  finite. 

By  means  of  these  values  of  a'  and  b'  we  may  simplify  tlie 
equation  (2)  as  follows:  Multiplying  the  first  of  equations  (a) 
by  a',  the  second  by  b\  and  adding,  we  have 

ma"  +  nb"  +  2Ia'b'  =  -  a'p  -  b'q.  (b) 

By  means  of  the  equations  (a)  and  (J)  the  general  equation 
(2),  omitting  accents,  is  reduced  to 

mx^  +  2lxy  +  nf  +  a'p  -^b'q^d^^,  (4) 

This  equation  (4)  will  now  represent  the  same  curve  as 
(1),  only  referred  to  new  axes  of  co-ordinates. 


GENERAL  EQUATION  OF  THE  SECOND  DEGREE.      203 

189.  A  second  fundamental  property  of  the  locus  of  the 
second  degree  is  immediately  deducible  from  (4).  If  x  and 
2/ be  any  values  of  the  co-ordinates  which  satisfy  this  equation, 
it  is  evident  that  —  x  and  —  y  will  also  satisfy  it.  That  is, 
if  the  point  {x,  y)  lie  on  the  curve,  the  point  (—  x,  —  y)  will 
also  lie  upon  it.  But  the  line  joining  these  two  points  passes 
through  the  origin  and  is  bisected  by  the  origin.  When  re- 
ferred to  the  original  system  (1),  this  origin  is  the  point 
whose  co-ordinates  are  a'  and  b'  in  (3).     Hence: 

Theorem  II.  For  every  curve  of  the  second  degree  there  is 
a  certain  point  ivhich  bisects  every  chord  of  the  curve  passing 
through  it. 

Def.  The  point  which  bisects  every  chord  passing  through 
it  is  called  the  centre  of  the  curve. 

Eemark  1.     In  the  special  case  when 

r  —  mn  =  0, 

the  centre  («',  i')  of  the  curve  will  be  at  infinity,  and  the 
theorem  will  not  be  directly  applicable. 

Remark  2.  Since  in  the  equation  (4)  the  origin  is  at  the 
centre,  this  equation  is  that  of  the  general  curve  of  the  second 
degree  referred  to  its  centre  as  the  origin. 

190.  Change  of  Direction  of  the  Axes  of  Co-ordinates. 
The  next  simplification  of  the  equation  will  consist  in  remov- 
ing the  term  in  xy.  To  do  this,  lot  us  refer  the  curve  to  the 
same  origin  as  in  (4),  namely,  the  centre,  but  to  a  new  system 
of  axes  making  an  angle  d  with  those  of  the  original  system. 
This  we  do  by  the  substitution  (§  27) 

X  =  x'  CO?,  S  —  y'  sin  S\ 
y  z=  x'  sin  ^  -\-  y'  cos  S. 

Making  this  substitution,  the  equation  becomes 
{m  co^^d  -\-  n  sin'd  +  2Z  %md  cos  d)x'^ 
-\-{m  sin'cJ  -}-  n  cos^(J  —  2Zsin  6  cos  cJ),?/'^ 
+  [  (?i  -  m)  sin  2d  +  2?  cos  ^d^x'if  =  d\ 

where  we  put        —  d'  =  a'p  +  b'q  -\-  d.  (5) 


204  PLANE  ANALYTIC  GEOMETRY. 

Substituting  for  the  powers  and  products  of  sines  and  co- 
sines their  values,  namely, 

cos'(^  =  ^(1  +  cos  26^), 
sin'(J  =  i(l  -  cos  2^), 
2  sin  S  coe6  =  sin  2^, 

and  then  putting,  for  brevity, 

li  =  {n  —  m)cos  2(y  —  21  sin  2(5",  )  /  n 

1c=\n-  m)sin  %6  +  2Z  cos  26,)  ^  ^ 

this  equation  reduces  to 

(m  -\-n-  h)x"  +(m  +  ^i  +  A)^/''  +  2kx'y'  =  2d\     (6) 

To  make  the  term  in  a;'^'  disappear,  we  must  so  determine 
the  value  of  6  that  ^  =  0.     This  gives 

tan  2d  = , 


m  —  n 


which  determines  the  values  of  d. 

Then  from  (c)  we  have,  when  k  =  0, 

h  sin  2(J  —  y^  cos  2d  =  -  2Z     =h  sin  2d; 
7i  cos  2S  -\-  Jc  sin  2d  =  ?i  —  w  =  A  cos  2d. 


The  values  of  h  and  d  are  therefore  given  by  the  equations 

h  sin  2d  =  - 
h  cos  2d  =  71 
whence 


sin  2d  =  -  2?;     )  .^. 


7i  =  >/(?w  -  ny  +  4/^  (8) 

Omitting  accents,  the  equation  (G)  of  the  curve  now  re- 
auces  to 

m-\-7i-  V{7n-nY-\-4:r  ,  ,  m  +  y^  +  ^{^i  -  nY-jTr  „ 

2d'  '^"^  2d'  r-i-(9) 

The  coefficients  of  x'  and?/'*  in  this  equation  are  always  real, 
but  may  be  either  positive  or  negative  according  to  the  sign  of 
d'  and  the  values  of  m,  n  and  I, 


OENERAL  EQUATION  OF  THE  SECOND  DEGREE.     205 
If,  then,  wc  put 


Id' 


m  +  ?i  -  V{m  -  ny  +  4.1' 
m  +  n+  ♦/(?«  -  n)'  +  il' 


(10) 


the   algebraic   signs  being  so  taken   that   a""  and   h'  without 
sign  shall  be  positive,  the  equation  (9)  still  further  reduces  to 

±5±i;=i,  (11) 

which  still  represents  the  same  locus  as  (1),  only  referred  to 
different  axes  and  a  different  origin. 

191.  There  are  now  three  cases  to  be  considered* 
Case  I.     The  algebraic  signs  in  the  first  member  both 

negative. 

Case  II.     The  algebraic  signs  both  positive. 

Case  III.     The  one  sign  positive  and  the  other  negative. 

In  the  first  case  the  equation  is  impossible  Avith  any  real 
values  of  x  and  y,  because  the  first  member  will  then  be  neces- 
sarily negative,  while  the  second  is  positive.  The  curve  is 
therefore  wholly  imaginary. 

In  the  second  case  the  equation  is  that  of  an  ellipse  whose 
semi-axes  are  a  and  h. 

In  the  third  case  the  equation  is  that  of  an  hyperbola  whose 
semi-axes  are  a  and  b. 

We  therefore  conclude: 

Theorem  III.  The  locus  of  the  eqiiation  of  the  second  de- 
gree between  rectangular  co-ordinates  is  a  conic  section. 

192.  Special  Kinds  of  Co7iic  Sections.  In  order  that  the 
equation  (9)  shall  represent  an  ellipse  we  must  have,  by  Case  II., 


m  -[-n>  V\m  —  nf  +  4^. 
Hence  {m  +  nfy  (m  -  ny  +  4:P 

and  mn  >  1% 

or  nm  —  V  positive. 


206  PLANE  ANALYTIC  GEOMETRY. 

Hence: 

Theorem  IV.  The  criterioii  whether  the  general  equation 
(1)  shall  represent  an  ellipse  or  an  hyperbola  is  given  hy  com- 
paring  the  square  of  the  coefficient  of  xy  ivith  four  times  the 
product  of  the  coefficieiits  of  x^  and  y^. 

If  the  square  is  algebraically  less  than  four  tiines  the  product, 
the  curve  is  an  ellipse  ;  if  greater,  it  is  an  hyperbola. 

In  special  cases  the  equation  may  represent  other  lines  than 
the  ellipse  or  hyperbola.  We  have,  in  fact,  tacitly  assumed 
that  the  expressions  a'  and  ¥  in  (10)  are  both  finite  and  de- 
terminate. We  have  now  to  consider  the  case  when  either  of 
them  is  zero,  infinity  or  indeterminate. 

193,  The  Parabola.  If  in  the  equation  (1)  mn  =  r, 
the  preceding  criterion  will  give  neither  a  genuine  ellipse  nor 
hyperbola,  but  a  limiting  curve  between  the  two.  We  know 
the  parabola  to  be  such  a  curve.  In  this  case,  also,  the  co-ordi- 
nates a'  and  b'  of  the  centre  of  the  curve  in  (3)  will  be  infinite, 
so  that  the  equation  cannot  be  reduced  to  the  form  (4).  But 
when  the  centre  of  an  ellipse  or  hyperbola  recedes  to  infinity, 
we  know  from  Elementary  Geometry  that  the  curve  becomes 
a  parabola.     We  shall  now  prove  this  result  analytically. 

Reduction  in  the  Case  of  a  Parabola,  We  have  to  con- 
sider the  special  form  of  the  general  equation  (1)  in  which 
I  =   Vmn.     The  equation  may  then  be  written  in  the  form 

{nfx  +  n^yY  -f  ^x  -^^y^d  =  0.  (12) 

That  is,  in  this  case  the  sum  of  the  three  terms  of  the  second 

1  1 

order  forms  the  square  of  the  linear  expression  m^x  -j-  n^y. 

AVe  may  infer  that  the  line  whose  equation  is 
nfx  -j-  n^y  =  0 

stands  in  some  special  relation  to  the  curve.  We  shall  there- 
fore so  change  the  direction  of  the  axes  of  co-ordinates  that 
this  line  shall  be  the  new  axis  of  X.  Taking  the  general 
equation  for  this  transformation, 

ic  =  cc'  cos  d  —  y'  sin  d,  \  /  x 

y  =  x'  sin  S  -\-  y'  cos  d,  \ 


OENEEAL  EQUATION  OF  THE  SECOND  DEGREE.     207 

we  see  that  they  give 

m  X  +  11^ y  =  {m^  cos  S  -\-  tf  sin  S)x' 

-\-(—  i)v  sin  d  -{-  w  cos  ^)y'.     {b) 

In  order,  now,  that  the  line  771  x  -j-  7iy  =  0  may  be  identi- 
cal witli  the  line  y'  =  0  (which  is  the  axis  of  X'),  the  coeffi- 
cient of  a;'  in  the  above  equation  must  vanish.  That  is,  we 
must  have 

??i  cos  d  +  n^sin  d  =  0,  (c) 

or  tan  d  = ;-; 

,  .     „  tan  S  mi 

whence  sin  o 


Vl  -f  tan'd  Vm  +  n 

cos  o  —  =  -4- 


Vl  -\-  tan^^d  V7n  +  7i' 

—  wi*  sin  (5'  -|-  ?t  cos  d  =  (m  +  ^)^  (^) 

(5)  and  (a)  now  become 

m  X  -\-  n^y  =  \/{m  -|-  ?^)?/'; 
nix'  -\-  7nh/'  n^y'  —  m^x' 

By  substitution,  the  equation  (12)  now  becomes 

and  putting,  for  brevity, 

(13) 


1  p 

_  qmi  —  pni 

X  -t 

~     {711  +  7l)l  ' 

iQ 

_p7ni  -\-  q7ii 
~  (771  +  ?^)§ ' 

D- 

d 

771  -\-  7l' 

the  equation  reduces  to 

7/'  +  Qy'  -  Fx'  +  D  =  0. 


208  PLANE  ANALYTIC  GEOMETRY. 

This,  again,  can  be  expressed  in  the  form 

(y'  +  kQY  =  W  +  P'-^'  -  D.  (15) 

We  can  still  further  simplify  this  equation  by  changing 
the  origin  to  the  point  whose   co-ordinates  are  —  ^Q  and 

^^ .     If  the  new  co-ordinates  referred  to  this  origin  are 

X  and  y,  we  have 

x  =  x'  -\ -p ; 

2/  =  /  +  iS. 
Then,  by  substitution,  the  equation  becomes 

y'  =  Px,  (16) 

which  is  the  equation  of  a  parabola  whose  parameter  is  j/P 
referred  to  its  vertex  and  principal  axis.  ^ 

We  therefore  conclude: 

Theorem  V.  The  general  equation  of  the  second  degree 
represents  a  imralola  luhen  the  square  of  the  coef[icient  of  xij 
is  equal  to  four  times  the  product  of  the  coefficient  ofx^  and  if. 

194.  Case  tvhen  the  Parameter  is  Zero.  There  is  still  a 
special  case  of  the  parabola  to  be  considered,  namely,  that  in 
which  P  =  0.  From  (16)  it  would  then  follow  that  y  =  0  for 
all  values  of  x.  But  this  conclusion  would  be  premature, 
because  the  transformation  (15)  would  then  involve  the  plac- 
ing of  a  new  origin  at  infinity.  We  must  therefore  go  back 
to  the  equation  (14),  which,  when  P  =  0,  gives 


y=-iQ±  ViQ'  -  D; 

that  is,  y  may  have  either  of  two  constant  values. 

Hence,  when  P  =  0,  the  equation  represents  a  pair  of 
straight  lines  parallel  to  the  axis  of  Xand  distant  V^Q'^  —  D 
on  each  side  of  it. 

195.   General  Case  of  a  Pair  of  Straight  Lines. 
On  reducing  to  the  form  (4),  the  absolute  term  d^  may 
vanish,     The  reduction  to  the  form  (9)  will  then  be  impossi- 


GENERAL  EQUATION  OF  THE  SECOND  DEGREE.      209 

ble,  because  the  coefficients  of  x^  and  y^  will  become  infinite. 
In  this  case,  however,  the  equation  (4)  will  bo 

mx^  +  2lxij  +  ny""  =  0.  (17) 

If  we  factor  this  quadratic  equation  by  any  of  the  metliods 
exphiined  in  Algebra,  we  may  reduce  it  to  the  form 


{ny  +  (^  +  yl'  -  'inn)x]  X  \ny  -\- {I  -  VP  -  m?i)x}  =  0, 

or  we  may  prove  this  equation  by  executing  the  indicated 
multiplications  and  thus  reducing  it  to  the  form  (17). 

Now  this  equation  may  be  satisfied  by  equating  either  of 
its  two  factors  to  zero.  If  we  distinguish  the  values  of  y  in 
the  two  factors  by  subscript  indices,  we  may  have  either 


or 


^  _  -  ^  +  ^/^ 

—  mn 

^^  -               2n 

-l-Vl' 
2/.  = 

—  mn 

X 

(18) 


that  is,  to  each  value  of  x  will  correspond  these  two  values  of 
y.  But  each  equation  (18)  is  that  of  a  straight  line  passing 
through  the  origin.     We  therefore  conclude: 

Theorem  VI.  When,  on  reducing  the  general  equation  of 
the  second  degree  to  the  centre,  the  absolute  term  vanishes, 
the  equation  represents  a  ])air  of  straight  lines. 

If  we  have  Z*  <  mn,  the  lines  will  both  be  imaginary. 
But  in  this  case  there  will  be  one  pair  of  real  values  of  the  co- 
ordinates, namely,  a;  =  0  and  y  =  0.     Hence, 

If,  in  the  case  supposed  in  the  preceding  theorem,  the  lines 
lecome  imaginary,  the  equation  can  he  satisfied  ly  only  a  single 
real  'point. 

This  result  is  also  evident  by  a  comparison  of  equations 
(9),  (10)  and  (11),  because  when  d'  =  0  and  P  <  7nn,  we 
have  an  ellipse  of  which  both  the  axes  are  zero,  and  this  can 
be  nothing  but  a  point. 

On  the  other  hand,  if  both  the  axes  of  an  hyperbola  be- 
come zero,  it  reduces  to  a  pair  of  straight  lines. 

We'  have    thus  found   two  seemingly  distinct  cases  in 


210  PLANE  ANALYTIC  GEOMETRY. 

which  the  conic  is  reduced  to  two  straight  lines:  the  one  when 
r  —  7nn  =  0  und  P  =  0;  the  other  when  d'  =  0.  We  shall 
now  show  that  the  former  cases  may  bo  combined  with  the 
latter. 

If  in  the  expression  (5)  for  d^  we  substitute  for  a'  and  b' 
their  values  (3),  it  will  become 

—  d'  =  i^(^^i^  ~  ^g)  +  g(^"g^  ~  ^P)  +  ^(^'  ~  ^^^^0 
r  —  7nn 

Now  let  us  put 

R  =p{np  —  Iq)  -f-  q{mq  —  Ip)  -\-  d(r  —  m7i),         (19) 

so  that  we  have 

R  =  d'{imi  -  r).  (20) 

If  we  square  the  value  (13)  of  P  and  note  that  we  are 
considering  the  case  when  mn  =  V,  we  have 
f  {m  +  nyP'  =  mq'  -  %lpq  +  nf 

^pi^wp  -  Iq)  +  q{mq  -  Ip). 
This  expression  is  zero,  by  hypothesis,  since  P  =  0.  Com- 
paring it  with  (19)  and  noting  thatZ^  --  nm  =  0,  we  see  that 
the  value  of  R  vanishes  in  this  case  as  it  does  when  d'  =  0. 
We  therefore  conclude  that  i2  =  0  is  the  condition  that  the 
conic  shall  reduce  to  a  pair  of  straight  lines. 

196.  Summary  of  Coiichisions.     The  various  conclusions 
which  we  have  reached  may  be  recapitulated  as  follows: 
The  general  equation  of  the  second  degree, 

rax"  +  2lxy  +  ny""  +  ^x  -\-  2qy  -\r  d  -  0, 

represents 

An  ellipse  when       r  <  mu] 
A  parabola  lohen     T  =  mn\ 
An  hyperbola  when  V  >  mn. 
Also,  in  spfecial  cases, 

The  ellipse  may  be  reduced  to  a  point; 
The  parabola  to  a  pair  of  parallel  straight  lines; 
The  hyperbola  to  a  pair  of  intersecting  straight  lines. 
But  since,  in  the  first  case,  the  point- ellipse  is  defined  as 
the  real  intersection  of  a  pair  of  imaginary  straight  lines,  we 


GENERAL  EQUATION  OP  THE  SECOND  DEGREE.     211 

may  describe  all  three  of  these  cases  as  one  in  which  the  conic 
is  reduced  to  a  pair  of  straight  lines,  and  sum  up  the  conclu- 
sion thus: 

If  the  coefficients  in  the  gc7ieral  equation  of  a  conic  satisfy 
the  condition 

p{np  -  Iq)  +  q(mq  —  Ijj)  +  d{r  -  mn)  =  0,       (19) 

the  conic  will  he  reduced  to  a  pair  of  straight  lines.  If  we 
have 

r  <  m7i,  the  lines  are  imaginary; 

r  =  7n7i,  the  lines  are  real,  and  parallel  or  coincide7it; 

r  >  mn,  the  lines  are  real  and  intersecting. 

19*7.  All  these  forms  are  conic  sections.  That  the 
ellipse,  parabola  and  hyperbola  are  such  sections  is  shown  in 
Geometry. 


When  the  cutting  plane  passes  through  the  vertex  of  the 
cone,  the  section  is  a  point  or  a  pair  of  intersecting  straight 
lines  according  to  the  position  of  the  plane. 


212  PLANE  ANALYTIC  OEOMETRY. 

When  the  vertex  of  the  cone  recedes  to  infinity,  the  base 
remaining  finite,  the  cone  becomes  a  cylinder,  and  the  sec- 
tion parallel  to  the  elements  is  a  pair  of  parallel  straight  lines. 

Remark.  A  conic  section  is,  for  brevity,  frequently 
called,  a  conic  simply,  and  we  therefore  designate  all  loci  of 
the  second  degree  as  conies. 

198,  Similar  Conies.  From  equation  (10)  it  follows 
that  the  ratio  a  :  b  of  the  semi-axes  depends  only  upon  the 
coefficients  m,  n  and  I  of  the  terms  of  the  second  order  in 
the  general  equation.     Since  we  have,  for  the  eccentricity, 

a 

it  follows  that  the  eccentricity  depends  only  on  the  same  co- 
efficients, I,  m  and  n. 

Moreover,  the  angle  d  which  the  principal  axes  of  the 
conic  form  with  the  original  axes  of  co  ordinates  depends  only 
on  these  same  coefficients.     Hence,  using  the  definitions. 

Similar  conies  are  those  which  have  the  same  eccen- 
tricity or  (which  is  the  same  thing)  the  same  ratio  of  the 
two  principal  axes;  Similar  conies  are  said  to  be  similarly 
placed  when  their  corresponding  axes  are  parallel, — we  have 
the  theorem: 

Theorem  VII.  All  conies  whose  equations  have  the  same 
terms  of  the  second  degree  vn  the  co-ordinates  are  similar 
and  similarly  placed. 

199.  Theorem  VIII.  A  conic  section  may  he  made  to 
pass  through  any  five  points  in  a  plane. 

Let  us  divide  the  general  equation  (1)  by  d,  and,  distin- 
guishing the  new  coefficients  by  accents,  we  have 

m'x'  +  n'y""  +  Uxy  +  2p'x  +  ^'y  +  1  =  0.        {a) 

Now  if  (a;,,  ?/,),  (ic,,  ?/J,  (2^3,  2/3),  (x^,  y^),  {x„  y,)  are  the 
five  given  points  in  the  plane,  we  have,  by  substituting  in  the 
last  equation  the  co-ordinates  of  these  five  points  for  the  gen- 
eral co-ordinates  x  and  y,  the  five  following  equations  of  con- 
dition: 


GENERAL  EQUATION  OF  THE  SECOND  DEGREE.    213 

m\'  +  oi'y^'  +  ^ZVx,y^  +  2p%  +  2q'y,  +  1  =  0; 
m'x,'  +  n>,^  +  2Z'.r,y,  +  2/;'a;,  +  2q'y,  +  1  =  0; 
m^x:  +  n'i/3»  +  2/^2/3  +  ^i^'^3  +  ^'y.  +  1  =  0; 
m^x:  +  7iV,»  +  n'x,y,  +  2i>X  +  ^'y.  +  1  =  0; 
7;iV  +  ^i'l/,^  +  %l\y,  +  2;;':^:,  +  2q'y,  +  1  =  0; 

from  which  the  coefficients  m',  n\  V,  'p'  and  q'  may  be  found, 
since  x^,  ;/„  a*^,  y^,  etc.,  are  hnoioii  quantities.  Substituting 
these  values  of  m',  n' ,  etc.,  in  the  general  equation  (a),  the 
resulting  equation  of  the  second  degree  in  x  and  y  will  be 
that  of  the  required  conic  section. 

Cor,  Since  the  equations  of  condition  are  all  of  the  first 
degree  with  respect  to  m* ,  n' ,  V,  y'  and  q' ,  each  of  these 
quantities  has  only  one  value;  tlierefore  only  one  conic  section 
can  be  passed  tJiroiigh  five  given  points  on  a  plane. 

Example.  Let  it  be  required  to  pass  a  conic  section 
through  the  five  points  (2,  1),  (-  1,  -  3),  (0,  3),  (1,  0), 
(3,  -  2). 

The  equations  of  condition  which  determine  the  coefficients  m,  n,  ?, 
etc.,  are  (omitting  the  accents) 


4m-{-    n-\-    4l-\-4p  + 

m  -}-  9/i  +    6^  —  2j)  — 

9/1                      + 

Qjn  +  4n  —  121  +  6p  - 

2?  + 1  =  0; 
6g  +  l  =  0; 
6g  -f  1  =  0; 
+  1=0; 
iq  +  1  =  0; 

>m  which  we  find 

33                       41                       1 
""64'        ^           192'                    32"' 

97 

^  -       128  ' 

59 
^=384 

Substituting  these  values  in  the  general  equation 

mx'^  +  n7f  +  2lxy  +  2px  +  2gy  +  1  =  0, 
and  clearing  of  fractions,  we  have 

99.?^2  _  41^3  _  i2xy  -  291a;  +  59y  +  192  =  0, 
which  is  the  equation  of  an  hyperbola,  since  I-  —  mn  is  a  positive  quantity. 

If  one  of  the  given  points  should  be  the  origin,  the  corresponding 
equation  would  be  the  impossible  one  1  =  0.  In  this  case  we  should 
have  to  divide  by  some  other  coefficient  than  d. 


214  PLANE  ANALYTIC  OEOMETRT. 

300.  Intersection  and  Tangency  of  Conies. 
Theorem    IX.     Tico  co7iics  in   general    intersect    each 

other  in  four  points. 

Proof.  The  co-ordinates  x  and  y  of  the  points  of  inter- 
section of  two  conies  are  given  by  the  roots  of  two  equations 
of  the  second  degree  in  x  and  y.  Now,  it  is  shown  in  Algebra 
that  when  we  eliminate  an  unknown  quantity  from  two  quad- 
ratic equations,  the  resulting  equation  in  the  other  unknown 
quantity  will,  in  general,  be  of  the  fourth  degree.  This  equa- 
tion will  therefore  have  four  roots,  thus  giving  rise  to  four  sets 
of  co-ordinates  of  the  points  of  intersection. 

Remark.  The  roots  may  be  all  four  real;  one  pair  real 
and  one  pair  imaginary;  or  all  four  imaginary;  and,  in  any 
case,  the  two  roots  of  a  pair  may  be  equal. 

According  as  this  happens  the  conies  are  said  to  intersect 
in  real,  imaginary  or  coincident  points.  In  the  latter  case 
they  are  said  to  touch  each  other  at  the  coincident  points. 

Cor.  Tivo  conies  may  touch  each  other  at  two  jwints  and 
no  more. 

301.  Families  of  Conies.     Let  us  put,  for  brevity, 
P'  =  mV   +  2Vxy  +  n'y''   +  2fx  +  ^'y   +  d'  ; 

etc.  etc.  etc. ; 

that  is,  let  us  represent  by  P',  P",  etc.,  any  functions  of  the 
second  degree  in  the  co-ordinates. 

Theorem  X.  If  P'  =  0  and  P"  =  0  are  the  equations 
of  any  two  different  conies,  then  the  equation 

}xP'  +  ;iP"  =  0  (20) 

{where  }i  and  \  are  constants)  tvill  represent  a  third  conic 
passing  through  the  four  points  of  intersection  of  the  other 
tivo. 

For,  first,  we  see  by  substitution  of  the  values  of  P'  and 
P"  that  the  equation  (20)  is  of  the  second  degree  in  the  co- 
ordinates.    Hence  its  locus  is  some  conic. 

Secondly,  every   pair   of  values  of  x  and  y  which  make 


GENERAL   EQUATION  OF  TUE  SECOND  DEGREE.     215 

both  P'  =  0  and  P"  =  0  must  also  satisfy  the  equation 
)uP'  4-  A.P"  =  0.  Hence  every  point  common  to  P'  and  P" 
must  belong  to  the  locus  of  (20);  that  is,  this  locus  passes 
through  all  the  points,  real  and  imaginary,  in  which  i^'  and 
P"  intersect.     The  number  of  these  points  is  four. 

By  giving  different  values  to  the  ratio  X  :  //,  any  number 
of  conies  passing  through  the  same  four  points  may  be  found. 
We  ma}^,  without  loss  of  generality,  suppose  //  =  1  in  this 
theory,  because  the  locus  (20)  depends  only  on  the  ratio  X  :  jn 

Def.  A  system  of  conies  all  of  which  pass  through  the 
same  four  points  is  called  a  family  of  conies. 

203.  Theorem  XL  In  a  family  of  conies  tiuo  and 
no  more  are  parabolas. 

Proof     If,  in  the  expression 

P   =:   P'  -\-  AP", 

we  substitute  for  P'  and  P"  their  values,  we  shall  have,  in  P, 

Coefficient  of  x^  =  m'  -\-  Xm"  =  m; 
Coefficient  of  ^  =  7i'  -f  '^^^^  =  ''^ 
Coefficient  of  2xy  =  V    ^  XI"   =  I 

The  condition  that  the  curve  P  =  0  ishall  be  a  parabola 
then  becomes 

0  =  r  —  mn 
=  (l"^  -  m!'n*')V-\-  {%VV'-  m'n"-  m!'n')X  +  Z"  -  mV. 

This  is  a  quadratic  equation  in  X,  which  therefore  gives 
two  values  of  A,  and  thus  two  expressions  for  P,  each  of  which, 
equated  to  zero,  is  the  equation  of  a  parabola.     Q.  E.  D. 

303.  Theorem  XII.  In  a  family  of  co7iics  three,  and 
no  more,  may  he  pairs  of  lines. 

Proof.  Forming  the  expression  P'  +  XP'\  we  find  the 
coefficients  of  the  general  equation  to  become 

m  =  m'  -\-  Am"; 

n  =  n'  -f  Xn"\ 

etc.  etc. 


216 


PLANE  ANALYTIC  GEOMETRY. 


In  order  that  a  conic  of  the  family  may  be  a  pair  of  lines 
it  is  necessary  and  sufficient  that  its  coefficients  satisfy  the 
condition  (19).  Each  term  of  (19)  is  of  the  third  degree  in 
the  coefficients.  Ilence  the  entire  condition  gives  an  equation 
of  the  third  degree  in  A,  which  has  three  roots.  Ilence  we 
have  three  expressions  of  the  form  F'  +  \P",  each  of  which, 
when  equated  to  zero,  gives  a  pair  of  lines.     Q.  E.  D. 

Remark.  If  we  call  A,  B,  C  and  D  the  four  points  of 
intersection  of  the  family,  the  three  pairs  of  lines  which  be- 
long to  it  will  pass  as  follows: 

One  pair  through  AB  and  CD  respectively; 
One  pair  through  ^(7  and  BD  respectively; 
One  pair  through  AD  and  ^(7  respectively; 

and  the  three  pairs  will  form  the  sides  and  diagonals  of  a 
quadrilateral. 


204.  Theorem  XIII.  If  loe  tahe  any  point  {x^,  y^) 
at  pleasure  in  the  plane  of  a  family  of  conies,  then  one  conic 
of  the  family,  and  no  more,  will  pass  through  this  point. 

Proof     Since  the  equation 

must  be  satisfied  for  the  value  {x^,  y^)  of  the  co-ordinates 
X  and  y  which  enter  into  it,  we  have 

(m'  +  Xm'')x^'  +  {n'  +  'Xn")y,'  +  2(^'  +  W)x,y,  +  etc.  =  0. 


GENERAL   EQUATION  OF  THE  SECOND  DEGREE.     217 

Since  x^,  ?/,  and  all  the  symbols  except  A.  in  this  equation 
are  known,  it  is  an  equation  of  the  first  degree  in  X  and 
so  has  but  one  root.     This  root  may  be  expressed  in  the  form 


in  which  P/  and  P/'  represent  the  values  of  P'  and  P"  when 
x^  and  y^  are  substituted  for  x  and  y.  There  being  but  one 
value  of  A,  only  one  conic  of  the  family  can  pass  through 
the  point  {x^,  ?/,). 

205.  THEOREii  XIV.     The  equation 

P'  4-  \p"  =  0  (a) 

may,  by  giving  all  real  values  to  A,  represent  every  possible 

conic  passing  through  the  four  i^iter sections  of  P'  and  P" , 

For,  let  G  be  any  conic  passing  through  the  four  points. 

P  ' 
Take  any  fifth  point  (x^,  y^  on  (7,  and  put  A,  =  —  -^^.     The 

equation  (a)  will  then  be  satisfied  identically  when  in  it  we  put 

x-x^,         y  =  y^, 

because  it  will  become 

P  ' 
P  '  —      ^    P  "  =  0 

Hence,  with  this  value  of  A,  (a)  will  represent  a  conic  of  the 
family  passing  through  the  point  (x^,  y^.  But  only  one 
conic  can  pass  through  five  points.  Hence  the  conic  thus 
found  will  be  C. 

306.  Relation  of  Focus  and  Directrix  to  the  General 
Equation.  Let  P^Cbe  any  conic  section;  GX,  OY,  rectan- 
gular axes.  Let  AQ,  the  axis  of  the  curve,  make  an  angle 
AGX=  a  with  the  axis  GJC.  And  let  (x,  y)  be  the  co-ordi- 
nates of  any  point  P;  {h,  h),  the  co-ordinates  of  the  focus  F\ 
and  r  =  GD,  the  distance  from  the  origin  to  the  directrix 
DK,     Join   PP,  and   draw  PE  perpendicular  to  DK,  and 


218 


PLANE  ANALYTIC  GEOMETRY. 


FH  i")arallel  to  OX.    Then,  by  the  definition  of  a  conic  section 
given  in  §  181,  Chapter  VIL,  we  have 

PF 

p-^  =  e,  the  eccentricity, 

and  therefore        PF  =  ePE 

and  PE  =  x  cos  a  -\-  y  ^m  a  —  r. 

Hence  PF  =  e{x  gos  a  -\-  y  sin  a  —  r), 

PF'  =  FH'  +  PH\ 
or     (x  cos  a  -{-  y  ^m  a  —  rye'  =  (x  —  h)'  -\-  (y  —  Icf. 

"Yl 


Expanding  and  collecting  terms,  we  have 

(1  —  e'  cos'^a:)^;'^  +  (1  —  e'  dn'a)y'  —  2e'  sin  a  cos  a  xy 
-\-  (2e'r  cos  a  —  2h)x  +  (J^e'i'  sin  a  —  2k)y 
+  {h'  +  F  -  e'r')  =  0.  (21) 

To  compare  this  with  the  general  equation,  we  must  divide 
both  it  and  the  general  equation  by  their  absolute  terms,  in 
order  that  the  two  may  have  the  same  coefificients.     Supposing 

171 

the  general  equation  thus  divided,  and  writing  7n  for  -r,  n  for 

-7,  etc. :  also  putting,  for  brevity, 

A  =  cos  a;  fx  =  sin  a; 


GENERAL  EQUATION  OF  THE  SECOND  DEGREE.     219 

we  find,  by  comparing  coefficients  of  corresponding  terms  in 
the  two  equations, 

1  -  e'X' 


or 


A'  +  F  -  eV  ~     ' 

1  -c'A»=m(F+;t' 

-cV); 

{a) 

1  -e'tx'  =  n{k-'  +  V 

-  eV); 

(i) 

-  e'XM  =  l{k'  +  V 

-  e'r'); 

(c) 

A'Ar  -h  =;j(F  +  /i" 

-  «V); 

id) 

e'nr  -h^  g{l:'  +  h' 

-  eV). 

ie) 

These  five  equations  completely  determine  the  five  quanti- 
ties a,  r,  h,  h  and  e,  and  hence  the  focus,  directrix  and 
eccentricity  of  the  conic,  in  terms  of  the  coefficients  of  the 
general  equation. 

EXERCISES. 

1.  Investigate  the  locus  represented  by  the  equation 

g 
Here  we  have        7n  =  4;       7i  =  l;       I  =  —. 

z 

Then  rnn- I''  =  4:  -  ^  = -\-\', 

4  4 

therefore  the  locus  is  an  ellipse. 

2.  Find  the  co-ordinates  of  the  centre  of  the  conic  repre- 
sented by 

bx"  +  ^'  +  2xy  -  Zlx  -  2y  -\-  100  =  0, 
and  find  the  angle  between  the  axis  of  the  curve  and  the  axis 
of  X. 

3.  "What  curve  does  y^  =  3{xy  —  2)  represent? 

4.  Determine  the  locus  y^  =  d{x  —  7)  and  the  angle  its 
axis  makes  with  the  axis  of  X. 

5.  Determine  the  locus  of  x^-\-  y^—  6xy  —  6x-\-2y  -\-6  =  0. 
Find  co-ordinates  of  the  centre,  and  the  angle  the  axis  of  the 
curve  makes  with  the  axis  of  X.  Aiis.  (0,  —  1);     135°. 

6.  If  Ay""  +  Bxy  -\-  Cx'  +  Dy  +  Bx -{-  F  =  0  be  the 
equation  of  a  conic  section,  show  that 

Bx  4-  2Ay  -{-  D  =  0 
is  the  equation  of  a  diameter  of  the  locus. 


220  PLANE  ANALYTIC  GEOMETRY. 

7.  From  the  equation  (9)  find  the  two  conditions  that  the 
equation  of  the  second  degree  shall  represent  a  circle. 

8.  Find  in  the  same  way  the  two  conditions  that  the  gen- 
eral equation  shall  represent  an  equilateral  hyperbola. 

9.  What  locus  is  represented  by  the  equation 

li'x''  -\-  mxy  -\-  k^'y'  —  c% 

when  m  =  lih'^ 

10.  Find  the  semi-parameter  of  the  parabola 

{x  -  yY  =  ax. 

11.  What  angle  do  the  asymptotes  of  the  hyperbola 

mx^  —  xy  =^  a 

make  with  the  transverse  axis? 

12.  If  we  have  the  two  conies 

mx""  -f  2lxy  +  ny''  +  ^px  -^  2qy  -^  d  =  0, 
mx^  +  ^Ixy  -\-  ny""  —  2px  —  2qy  -{-  d  =  0, 

show  that  the  line  joining  their  centres  is  bisected  by  the 
origin. 

13.  The  co-ordinates  x  and  ?/  of  a  moving  point  are  ex- 
pressed in  terms  of  the  time  t  by  the  equations 

X  =  mt  -}-  a;        y  =  mt  -\-  b. 

What  is  the  equation  of  the  line  described  by  the  point? 

14.  If  the  co-ordinates  are  given  by  the  equations 

X  =  mt,        y  =  nt', 

show  that  the  curve  is  a  parabola,  and  express  its  parameter. 

15.  What  condition  must  the  coefficients  of  the  general 
equation  (1)  of  the  second  degree  satisfy  that  the  curve  may 
pass  through  the  origin  of  co-ordinates? 

16.  Write  the  equation  of  that  conic  formed  of  a  pair  of 
straight  lines  through  the  origin  whose  slopes  are  m  and  —  m. 

17.  Do  the  same  thing  when  the  lines  are  to  intersect  in 
the  point  (a,  h). 

18.  What  is  the  condition  that  the  principal  axes  of  a 
conic  shall  be  parallel  to  the  axes  of  co-ordinates?    (See  §190. ) 


GENERAL  EQUATION  OF  THE  SECOND  DEGREE.     221 

19.  Express  the  points  in  which  the  locus  of  the  equation 

x"  -  2:c?/  +  ?/'  +  3a;  -  4  =  0 

cuts  the  respective  axes  of  co-ordinates. 

Ans,  The  axis  of  X,  (1,  0)  and  (-  4,  0); 
The  axis  of  Y,  (0,  2),  (0,  -  2). 

20.  What  condition  must  the  coefficients  of  (1)  satisfy  that 
the  curve  may  be  tangent  to  the  axis  of  X  and  to  the  axis  of 
Irrespectively? 

Ans.  p^  =  md  for  the  axis  of  X; 
q^  =  nd  for  the  axis  of  Y, 

The  solution  is  very  simple,  if  it  is  remembered  that  the  curve  is  to 
cut  the  axis  in  two  coincident  points. 

21.  Find  the  equation  of  that  conic  which  cuts  the  axis 
of  X  at  points  whose  abscissas  are  —  2  and  +  4,  the  axis  of 
l^at  points  whose  ordinates  are  —  1  and  +  2,  and  whose  princi- 
pal axes  are  parallel  to  the  axes  of  co-ordinates. 

Ans.  x"  +  4:y'^  —  2a;  —  4?/  —  8  =  0. 

22.  Show  that  in  the  general  equation  (1)  the  line 

mx  -}-  ly  —  p  =  0 

bisects  all  chords  parallel  to  the  axis  of  X.     Find  also  the  line 
which  bisects  all  chords  parallel  to  the  axis  of  Y. 

Begin  by  solving  the  general  equation  as  a  quadratic  in  x  so  as  to  ex- 
press X  in  terms  of  p,  and  vice  versa. 

23.  How  many  points  are  necessary  to  determine  a  para- 
bola?    An  equilateral  hyperbola? 

24.  Mark  five  points  at  pleasure  on  a  piece  of  paper. 
Can  you  find  any  criterion  for  distinguishing  at  sight  the  fol- 
lowing cases? — 

I.  The  five  points  lie  on  one  branch  of  a  conic  (ellipse  or 
hyperbola). 

II.  The  conic  is  an  hyperbola  having  three  of  the  points 
on  one  branch  and  two  on  the  other. 

III.  It  is  an  hyperbola  having  four  points  on  one  brancli 
and  one  on  the  other. 

Suppose  a  string  drawn  tightly  around  all  the  points,  and 
note  the  number  of  points  the  string  will  not  reach. 


222  PLANE  ANALYTIC  GEOMETRY. 

25.  Find  the  equation  of  a  parabola  which  shall  touch  the 
axis  of  X  at  the  point  whose  abscissa  is  +  ^>  ^^^  the  axis  of 
JTat  the  point  whose  ordinate  is  +  1- 

Alls,  x^  —  4:xy  +  ^y^  —  4a;  —  8y  -{-  4,  =  0. 

26.  The  base  of  a  triangle  has  a  fixed  length,  and  the 
escribed  circle  below  this  base  is  required  to  touch  it  at  a 
fixed  point.  Find  the  locus  of  the  point  of  intersection  of  the 
two  sides  of  the  triangle. 

27.  A  line  passes  through  the  fixed  point  (0,  i)  on  the 
axis  of  Y  and  intersects  the  axis  of  X  and  the  fixed  line 
y  =  mx.  Find  the  locus  of  the  middle  point  of  the  segment 
of  the  line  contained  between  the  fixed  line  and  the  axis  of  X. 

28.  Investigate  the  locus  of  the  point  the  differences  of 
the  squares  of  whose  distances  from  the  axis  of  X  and  from 
the  line  y  =  mx  is  the  constant  quantity  ^^ 

29.  The  base  <^  of  a  triangle  and  the  sum  of  the  angles  at 
the  two  ends  of  the  base  are  both  constant.  Investigate  the 
locus  of  its  vertex. 

30.  Each  abscissa  of  the  circle  x"^  -\-  y^  =  r^  is  increased  by 
m  times  its  ordinate.  Find  the  locus  of  the  ends  of  the  lines 
thus  formed. 

31.  Investigate  the  locus  of  the  middle  points  of  all  chords 
of  an  ellipse  which  pass  through  a  fixed  point. 

32.  The  circle  x^  +  2/'  =  ^*  has  two  tangents  intersecting 
in  a  movable  point  P  and  cutting  out  a  fixed  length  a  from 
a  third  tangent  y  =  r.     Investigate  the  locus  of  P. 

33.  Show  that  the  equation  of  that  pair  of  straight  lines 
formed  of  the  axes  of  co-ordinates  is  xy  =  0. 


PART    II. 
GEOMETRY  OF    THREE  DIMENSIONS. 


CHAPTER  I. 

POSITION  AND  DIRECTION  IN  SPACE. 


207.  Directions  and  Angles  in  Space.  Two  straight  lines 
cannot  form  an  angle,  as  that  term  is  defined  in  elementary 
geometry,  unless  they  intersect.  Two  lines  in  space  will,  in 
general,  pass  each  other  witliout  intersecting.  Hence  we 
cannot  speak  of  the  angle  between  such  lines  unless  we  extend 
the  meaning  of  the  word  angle.  Now  the  following  theorem 
is  known  from  solid  geometry: 

If  ive  have  given  any  two  lines,  a  and  h,  in  space; 

and  if  ive  take  any  point  P  at  pleasure; 

and  if  through  P  loe  draw  tioo  lines,  PA  arid  PB,  par- 
allel to  a  and  h  respectively,  — 

then,  so  lo7ig  as  we  leave  a  and  h  unchanged,  the  angle  AP B 
will  have  the  same  value  no  matter  where  ive  take  the  point  P. 

"We  therefore  take  this  angle  as  the  measure  of  the  angle 
between  the  lines  a  and  h.  This  measure  may  be  considered 
as  expressing  the  difference  of  direction  between  the  lines 
a  and  h,  and  the  word  angle,  as  applied  to  two  non-intersecting 
lines,  will  be  understood  to  mean  their  difference  of  direction. 
We  thus  have  the  following  definition  and  corollary: 

Def  Tlie  angle  between  two  non-intersecting  lines  is 
measured  by  the  angle  between  any  two  intersecting  lines 
parallel  to  them. 

Cor.  If  we  have  two  systems  of  parallel  lines  in  space,  the 
07ie  a,  a* ,  «",  etc.,  the  other  h,  h',  h",  etc.,  then  the  angles  hctiveen 
any  line  of  a  and  any  line  of  b  will  all  be  equal  to  each  other. 


224  GEOMETRY  OF  THREE  DIMENSIONS. 

208.  Projections  of  Lines.  The  projection  of  a  finite 
line  PQ  upon  an  indefinite  line  Xis  the  lengtli  oi  X  inter- 
cei)ted  by  the  perpendiculars  dropped  upon  it  from  the  two 
ends  of  PQ. 

Theorem  I.  The  projection  of  a  line  is  eqnal  to  the  2Jro- 
duct  of  its  length  by  the  cosine  of  the  angle  ichich  it  forms 
with  the  line  on  lohich  it  is  projected. 

To  prove  the  theorem,  pass  through  each  of  the  termini, 
P  aud  Q,  a  plane  perpendicular  to  JT.*  The  planes  will  be 
parallel  to  each  other  and  will  cut  X  at  the  termini  of  the 
projection  of  PQ,  which  we  may  call  P'  and  Q'.  Through 
P'  draw  P'Q'^  \\  PQ  and  intersecting  the  plane  through  Q 
in  Q".     We  shall  then  have 

P'e"  =  PQ', 

(being  parallels  between  parallel  planes;) 

P'Q'   ^  P'Q"  cos  Q"P'Q' 

=  PQ  cos  (angle  between  P'Q'  and  PQ).     Q.E.D. 

Remark.  By  assigning  a  positive  and  negative  direction 
to  the  two  lines,  the  algebraic  sign  of  the  projection  will  be 
determined.  It  will  be  positive  or  negative  according  as  the 
angle  between  the  positive  directions  of  the  two  lines  is  less 
or  greater  than  a  right  angle.  The  following  theorem  is  a 
result  of  this  convention,  combined  with  the  principles  of 
Trigonometry: 

Theorem  II.  If  we  have  any  broken  line  in  space,  made 
up  of  the  consecutive  straight  lines  AB,  BG,  CD,  etc.,  .  .  .  GH, 
■which  lines  form  the  angles  a,  ft,  y,  etc.,  tvith  the  line  of  i^ro- 
jection  X; 

and  if  we  project  this  line  upon  X  by  dropping  perpendicu- 
lars A  A\BB',  GG',  etc.,  .  .  .  .  HH',  meeting  X  at  the  points 
A',  B',  C',  D',  etc.,  ....  H'— 

then  the   length  A'FP  zvill  be   the   algebraic  sum   of  the 

*  No  figure  is  drawn  for  this  demonstration,  because  two  non-inter- 
secting lines  in  space  cannot  be  represented  on  paper.  If  the  student 
cannot  readily  conceive  the  relation,  he  should  take  two  rods  or  pencils 
to  represent  the  lines. 


POSITION  AND  DIRECTION  IN  SPACE. 


225 


separate  lengths  A'B\  B' C ,  CD' ,  etc.,  these  separate  lengths 
being  considered  j^ositive  when  taken  m  one  direction,  7iegative 
when  taken  in  the  oppodte  direction,  and  will  be  expressed 
by  the  equation 

A'H'  =  AB  cos  a -{-  BC  cos  /3-\-  CD  cos  y  +  etc. 

209.  Co-ordinate  Axes  and  Flaiies  in  Space.  The  position 
of  a  point  ill  space  maybe  defined  by  its  relation  to  three  straight 
lines  intersecting-  in  the  same  point  and  not  lying  in  a  plane. 

Three  such  lines  of  reference  are  called  a  system  of  co- 
ordinate axes  in  space. 


The  point  in  which  the  axes  intersect  is  called  the  origin 
of  co-ordinates,  or  simply  the  origin. 

The  three  axes  are  designated  by  the  letters  JT,  Y  and  Z 
respectively. 

The  co-ordinate  axes,  taken  two  and  two,  lie  in  three  planes, 
one  containing  the  two  axes  X  and  Y,  another  l^and  Z,  a 
third  Z  and  X. 

These  planes  are  called  co-ordinate  planes.  They 
are  distinguished  as  the  plane  of  XY,  the  plane  of  YZ,  and 
the  plane  of  ^X  respectively. 

The  several  angles  which  the  axes  of  co-ordinates  make 
witli  each  other  are  arbitrary.  But,  for  elementary  purposes, 
it  is  most  convenient  to  suppose  each  to  form  a  right  angle  with 


226  GEOMETRY  OF  THREE  DIMENSIONS. 

the  other  two.     The  following  conclusions  then  result  from 
solid  geometry: 

I.  Each  axis  is  perpendicular  to  the  plane  of  the  other  two. 

II.  Each  plane  is  perpendicular  to  the  other  two  planes. 

III.  Every  line  or  plane  perpendicular  to  one  of  the  planes 
is  parallel  to  the  axis  which  does  not  lie  in  that  plane. 

IV.  Every  line  or  plane  perpendicular  to  one  of  the  axes 
is  parallel  to  the  plane  of  the  other  two. 

V.  If  the  centre  of  a  sphere  lies  in  the  origin,  the  intersec- 
tions of  the  co-ordinate  axes  and  planes  with  its  surface  form  the 
vertices  and  sides  of  eight  trirectangular  spherical  triangles. 

VI.  To  each  plane  corresponds  the  axis  perpendicular  to 
it,  which  is  therefore  called  the  axis  of  the  plane. 

210.   Co-ordinates,    The  position  of  a  point  in  space  is 
defined  by  its  distances  from  the  three  co-ordinate  planes  of  a 
system,  each  distance  being  measured  on  a  line  parallel  to  the 
axis  of  the  plane.   When  the  axes  are  rectangular,  these  direc- 
tions will  be  perpendicular  to  the  planes.     The  notation  is: 
X  =  distance  from  plane  YZ] 
y  =  distance  from  plane  ZX; 
z  =  distance  from  plane  XT. 

To  distinguish  between  equal  distances  on  the  two  sides 
of  a  plane,  distances  on  one  side  are  considered  positive, 
on  the  other  negative. 

The  positive  direction  from  jeach  plane  is  the  positive  di- 
rection of  the  axis  perpendicular  to  it. 

It  is,  of  course,  a  matter  of  convention  which  side  we  take 
as  positive  and  which  negative.  A  certain  relation  between 
the  positive  directions  is,  however,  adopted  in  ph3^sics  and 
astronomy,  and  should  be  adhered  to.     It  is  this: 

The  positive  side  of  the  plane  of  XY  is    Y 
that  from  lohich  we  must  looh  in  order  that 
the  axis  OX  would  have  to  turn  in  a  direc- 
tion the  opposite  of  that  of  the  hands  of  a 
watch  in  order  to  talce  the  positio7i  OY. 

If  we  conceive  the  plane  of  XY  to  be  o' ^X 


horizontal,  the  axis  of  Z  will  be  vertical,  and,  supposing  the 


POSITION  AND  DIRECTION  IN  SPACE.  227 

axes  of  X  and  Y  to  be  arranged  as  in  plane  geometry,  the 
positive  side  of  tlie  plane  will  be  the  upper  one.* 

The  following  propositions  respecting  certain  relations  of 
signs  of  co-ordinates  to  position  should  be  perfectly  clear  to 
the  student : 

I.  The  co-ordinate  planes  divide  the  space  surrounding  the 
origin  into  eight  regions,  distinguished  by  the  distribution 
of  +  and  —  signs  among  the  co-ordinates. 

Imagine  the  axis  of  X  to  go  out  positively  toward  the  east; 
Imagine  the  axis  of  I^to  go  out  positively  toward  the  north; 
Imagine  the  axis  of  -^to  go  out  positively  upward, 

and  the  point  of  reference  to  be  the  origin.     Then — 

II.  For  all  points  above  the  horizon  z  will  be  positive;  for 
all  points  below  it,  negative. 

III.  For  all  points  east  of  the  north  and  south  line  x  will 
be  positive;  for  all  points  west  of  it,  negative. 

IV.  For  all  points  north  of  the  east  and  west  line  y  will  be 
positive;  for  all  points  south  of  it,  negative. 

311.  How  the  Co-ordinates  define  Position.  Let  us  first 
suppose  that  the  only  information  given  us  respecting  the 
position  of  a  point  P  is  its  co-ordinate 

X  —  a,  (a) 

a  being  a  given  quantity. 

This  is  the  same  as  saying  that  P  is  at  a  distance  a  from 
the  plane  YZ.  In  order  that  a  point  may  be  at  a  distance  a 
from  a  plane,  it  is  necessary  and  sufficient  that  it  lie  in  a  par- 
allel plane,  such  tiiat  the  distance  between  the  two  planes  is  a. 

Hence  the  proposition  informs  us  that  P  lies  in  a  certain 
plane. 

*  The  author  regards  it  as  unfortunate  that  many  mathematical 
writers,  in  treating  of  analytic  geometry,  reverse  the  arrangement  of  axes 
in  space  universally  adopted  in  astronomy  and  physics.  Uniformity  in 
this  respect  is  so  desirable  that  he  has  not  hesitated  to  adhere  to  the  latter 
arrangement. 

It  may  be  remarked  that,  in  drawing  figures,  the  axes  are  represented 
as  seen  from  different  stand-points  in  different  problems,  the  best  point 
of  view  for  each  individual  problem  being  chosen. 


228  OE0METE7  OF  THREE  DIMENSIONS. 

If  we  are  informed  that 

y  =  ^, 
then  P  must  lie  in  a  plane  parallel  to  ZX,  at  the  distance  1) 
from  it. 

If  both  propositions,  x  =  a  and  y  =  b,  are  true,  then  P 
must  lie  in  both  planes.  Hence  it  must  lie  on  their  line  of 
intersection,  which  line  will  be 

parallel  to  the  axis  of  Z, 
parallel  to  the  planes  ZX  and  YZ, 
and  perpendicular  to  plane  XY. 

If  it  is  also  added  that 

2;  =  c, 

the  point  P  lies  in  a  third  plane,  parallel  to  XY,  Lying  in 
all  three  planes,  its  only  position  will  be  their  common  point 
of  intersection. 

Hence: 

Tlie  position  of  a  point  is  completely  determined  luhen  its 
three  co-ordinates  are  given. 

NoTATioi^.  By  point  {a,  h,  c)  we  mean  the  point  for 
which  X  =  a,  y  =  h  and  z  =  c, 

212.  Paralleloinpedon  formed  hy  the  Co-ordinates. 

Let  0  be  the  origin;  OX,  OY,  OZ,  the  axes;  P,  the  point; 
PR,  PS  and  PQ,  parallels  to  the  axes  terminating  in  the  sev- 
eral planes.     Then,  by  definition,  the  co-ordinates  of  P  will 

be 

x  =  EP  =  VS  =  OT  =  WQ; 

y=:  SP  =  VP=  0W=  TQ; 

z  =  QP  =  TS  =  OV  =  WR. 

We  shall  then  have,  by  considering  the  three  planes  which 

contain  these  co-ordinates. 

Plane  EPS  ||  plane  XY; 
Plane  SPQ  \\  plane  YZ; 
Plane  QPR  \\  plane  ZX. 

Hence  the  three  planes  which  contain  the  co-ordinates, 
together  with  those  which  contain  the  axes,  form  the  six  faces 
of  a  parallelopipedon.     This  figure  has 


POSITION  AND  DIRECTION  IN  SPACE. 


229 


Four  edges  =  and 
Four  edges  =  and 
Four  edges  =  and 


to  co-ordinate  x', 
to  co-ordinate  y\ 
to  co-ordinate  z. 


X 


213.  Since  there  are  four  equal  lines  for  each  co-ordinate, 
we  may  use  any  one  of  these  four  in  constructing  the  co-ordinate. 
Sometimes  it  is  advantageous  to  choose  such  lines  that,  taking 
the  co-ordinates  in  some  order,  the  end  of  each  shall  coincide 


with  the  beginning  of  the  next  following,  the  end  of  the  third 
being  the  point.     For  example,  we  may  take,  in  order, 

X  =  OT; 
y  =  TQ; 
z  =  QF. 


230  GEOMETRY  OF  THREE  DIMENSIONS. 

The  three  co-ordinates  will  then  form  a  series  of  three  lines, 
each  at  a  right  angle  with  the  other  two. 

Again,  each  face  of  the  parallelopipedon  being  perpendicu- 
lar to  four  edges,  it  follows  that  the  diagonal  PT  will  be  the 
perpendicular  from  P  upon  the  axis  of  X,  and  that  the  like 
proposition  will  be  true  for  the  other  axes.     Hence 

The  rectangular  co-ordinates  of  a  point  are  equal  to  the 
segments  of  the  axes  contained  betioeen  the  origin  and  the  per- 
pendiculars dropped  from  the  point  upon  the  respective  axes. 

EXERCISES. 

1.  If  from  the  point  {a,  h,  c)  we  draw  lines  to  the  several 
points  {—  a,  b,  c),  (a,  —  b,  c),  {a,  b,  —  c),  (—  a,  —  b,  c), 
(a,  —  b,  —  c),  {—  a,  b,  —  c),  {—  a,  —  J,  —  c),  define  in  what 
seven  points  these  lines  will  cut  such  of  the  co-ordinate 
planes  as  they  intersect. 

2.  If  perpendiculars  be  dropped  from  a  point  {a,  b,  c)  upon 
the  three  co-ordinate  axes,  show  that  the  lengths  of  the  per- 
pendiculars will  be   Vd'  +  b\   Vb'  +  c'  and  Vc'  +  a\ 

3.  If  we  take,  on  each  axis,  a  point  at  the  distance  r  from 
the  origin,  what  will  be  the  mutual  distances  of  the  three 
points  from  each  other,  and  of  what  figure  will  they  and  the 
origin  form  the  vertices ? 

^sZt'-jL    ^'  ^^'  ^^^  ^^^®  ^^^^  ^^  ^'  1^  and  Z  respectively,  we  take 
the  points  P,  Q  and  R,  and  from  the  origin  0  drop  OLL  QR, 
i--^^-*-^MlRP  and  ONLPQ,  show  that 

or  ^  OM'  ^  ON'       OP'  ^  OQ'  ^  OR' 

214.  Problem.     To  find  the  distance  of  a  point  (x,  y,  z) 
from  the  origin,  and  the  a7igles  ichich  the  line  joining  it  to  the 
origin  makes  icith  the  co-ordinate  axes. 
Let  P  be  the  point,  and  let  us  put 

r,  the  distance  OP  from  the  origin; 
a,  y5,  r,  the  angles  POX,  POY and  POZ 
which  the  line  makes  with  the  axes. 
Then— 
I.  Because  OP  is  the  diagonal  of  a  rectangular  parallelo- 


POSITION  AND  DIRECTION  IN  SPACE.  231 

pipedon  whose  edges  are  PQ,   PR  and  PSy  we  have,  by 
Geometry, 

OP'  =  PQ'  4-  PR'  +  PS'; 
that  is, 

r'  =  x'-\-  y'  +  z'  (1) 

and  r  =  Vx'  -j-  y'  ~\-  z', 

which  gives  the  distance  of  the  point  from  the  origin. 

II.  Again,  supposing  the  same  construction  as  in  §  212,  we 
have 

pro  =  a  right  angle. 
Hence 

0T=  OP  COS  POX, 

or,  from  the  equality  of  the  parallel  edges, 

a:  =  r  cos  a. 
In  the  same  way  i  y  x 

y  =  r  cos  J3;  «  ^  ' 

z  =  r  cos  y. 

The  required  values  of  the  cosines  of  the  angles  are,  there- 
fore, 

X  X  ^ 

cos  a  =  —  =  — ; 


r        V,c^  -{-f-\-z^ 

z  z 

cos  ;/  =  -  = 


^         Vx""  +  y''  +  z"" 


(3) 


Theorem  III.  Tlie  sum  of  the  squares  of  the  cosines  of 
the  angles  ichich  a  line  through  the  origin  makes  with  three 
rectangular  axes  is  unity. 

Proof  Adding  the  squares  of  the  last  set  of  equations, 
we  have 

cos'a  +  cos'/?  +  oosV  =  J^—J  =  1,  (i) 

which  proves  the  theorem. 

This  theorem  enables  us  to  find  any  one  of  the  angles  a, 
/?  and  y  when  the  other  two  are  given. 


232 


GEOMETliT  OF  THREE  DIMENSIONS. 


215,  Problem.  To  express  the  distance  betioeen  two 
points  given  hy  their  co-ordinates,  and  the  angles  which  the 
line  joining  them  forms  loith  the  axes  of  co-ordinates. 

Let  P  and  F'  be  the  points,  and  let 

P   have  the  co-ordinates  x,    y,    z\ 


P'  have  the  co-ordinates  x\ 


y 


I* 


IP 


■>ivr 


Through  each  of  the  points  P  and  P'  pass  three  planes 
parallel  to  the  co-ordinate  planes.  These  planes  will  form  the 
faces  of  a  rectangular  parallelepiped  of  which  the  edges  are 

P'M  =  x'  -x\ 
P'N  =  y'  -y, 

P'R    ^    ^'   _   2. 

If  we  put 
A  =  PP\  the  distance  of  the  points,  we  hav3,  by  Geometry, 
A"  =  P'W  +  P'N''  +  P'R 

=  {x'  -  xy  +  0/'  -  yy  +  (.'  -  zy 

=  x'^  +  y''  +  ^"  +  ^^  +  f-\-  z'  -  Kxx'  +  yy'^r  zz^). 
Hence      J  =  V(x''^^^xy~~-f~{y'~^y'-\-  (z'  -  zy,  (3) 

To  express  the  angles  a,  j3  and  y  which  the  line  PP' 
forms  with  the  axes,  we  note  that  these  angles  are,  by  §  207, 
equal  to  those  which  P'P  forms  with  P'M,  P'N  and  P' R 
respectively.     Thus  we  find 


cos  a 


X'  —  X 


cos  ft  =  '^-j^-\  ^ 

z'  —  z 
COS  y  =  —-^-. 


(4) 


POSITION  AND  DIRECTION  IN  SPACE.  233 

216.  Problem.  To  express  the  angle  hetiveen  two  lines 
in  terms  of  the  angles  which  each  of  them  forms  with  the  co- 
ordinate axes. 

Let  the  two  lines  emanate  from  the  origin,  and  put 

a,  (3,  y,  the  angles  which  one  line  makes  with  tlie  axes; 
a' ,  yS',  y' ,  the  corresponding  angles  for  the  other  line. 

On  each  line  take  a  point,  namely,  P  on  the  one  and  P' 
on  the  other,  and  put 

r,  r' ,  the  distances  of  P  and  P'  respectively  from  the  origin. 

The  problem  is  solved  by  expressing  the  distance  PP'  \\\ 
two  ways: 

I.  The  equations  («)  of  §  214  give,  for  the  co-ordinates 

oi  P'.^x    —  r   cos  a\      y    —  r   cos  /?;      z    —  r   cos  y\ 
of  P'\  x'  —  t'  cos  a'\     y'  —  r'  cos  yS';     z'  —  r'  cos  /'. 

Substituting  these  values  in  the  expression  of  §  215  for 
the  distance  of  the  points, 

A-  =  7*'^(cos*<a:  +  cos''/?    +  cos';/) 
+  r'XcosV'  +  cos'^/J'  +  cosV) 
—  2rr'(cos  a  cos  a'  -\-  cos  y5  cos  /?'  +  cos  y  cos  y'). 

The  first  two  terms  reduce  to  r^  +  r"  by  {h). 

II.  If  we  put 

V  =  the  angle  between  the  lines, 

the  lines  r,  r'  and  A  will  be  the  sides  of  a  plane  triangle  of 
which  the  angle  opposite  the  side  A  is  v.  Hence,  by  Trigo- 
nometry, 

z/'  =  r'  -f-  ?'"  —  %rr'  cos  v. 

III.  Comparing  the  two  values  of  A"^,  we  find 

cos  i;  =  cos  <af  cos  a'  +  cos  /?  cos  ^'  +  cos  y  cos  y' ,    (5) 

which  is  the  required  expression. 

Cor.     The  condition  that  v  shall  be  a  right  angle  is 

cos  a  cos  a'  +  cos  ^  cos  yS'  +  cos  y  cos  y'  —  0,       (G) 

because,  in  oi-der  that  an  angle  shall  be  a  right  angle,  it  is 
necessary  and  sufficient  that  its  cosine  shall  be  zero. 


234  GEOMETRY  OF  TUliEE  DIMENSIONS. 

317.  Def.  The  direction-cosines  of  a  line  are  the 
cosines  of  the  three  angles  which  it  forms  with  the  co-ordi- 
nate axes. 

The  direction-cosines  are  so  called  because  they  determine 
the  direction  of  the  line. 

Direction-  Vectors.  The  three  direction-cosines  of  a  line 
are  not  independent,  because,  when  any  two  are  given,  the 
third  may  be  found  by  the  equation 

cos^a  +  cos^/?  -|-  cosV  =  1-  C^) 

But  the  direction  of  the  line  may  he  defined  ly  any  three 
quantities  proportional  to  its  direct ion-cosi7ies.  To  show  this, 
let  us  put 

I,  m,  n,  any  three  quantities  proportional  to  cos  a,  cos  /?, 
cos  y  respectively. 

Because  of  this  proportionality,  we  shall  have 
?      _     m     _     w     _     ^ 

cos  a      cos  fi  ~  cos  y  ~ 
whence 

a  cos  a  =  /;        o"  cos  /?  =  m;        a  cos  y  =  n. 
The  sum  of  the  squares  of  these  equations  gives,  by  (7), 
(?»  =  r  -f  m=  +  n\ 
I  I 


cos  a  = 


cos  /3 


a 

Vf 

-f  m^ 

+ 

n"" 

m 

= 

m 

a 

Vf 

+  w^ 

+ 

n^ 

n 



n 

(8) 


cos  r  —  -        , 

(?        |//2  _|.  ^yi"  +  n"" 

Thus,  when  I,  m  and  w  are  given,  the  angles  a,  /?  and  y 
can  be  found,  and  thus  the  direction  of  the  line  is  fixed. 

The  quantities  I,  m  and  n  are  called  direction-vectors. 

The  direction  of  the  line  depends  only  upon  the  mutual 
ratios  of  the  direction-vectors,  and  not  upon  their  absolute 
values.  For,  if  we  multiply  the  three  quantities  Z,  m  and  n 
by  any  factor  p,  a  will  be  multiplied  by  this  same  factor, 
which  will  divide  out  from  the  equations  (8),  and  thus  leave 
the  values  of  a,  /?  and  y  unchanged. 


POSITION  AND  DIRECTION  IN  SPACE.  235 

Cor,  If  the  directions  of  two  lines  are  given  by  the 
direction-vectors 

I,  m  and  w,         V,  m'  and  n' , 
respectively,    the    condition   that   they   shall    form   a   right 
angle  is 

IV  4-  mm'  +  nn'  -  0.  (9) 

For,  by  substituting  in  tlie  equation  (6)  the  values  of  the 
direction-cosines  of  the  two  lines  given  by  (8),  the  condition 
becomes 

IV  +  mm'  +  nn'  _ 

6o'  ~^' 

from  which  (9)  immediately  follows. 

218.  Problem.     To  exiwess  the  square  of  the  sine  of  the 
angle  hetiveen  tivo  lines  in  terms  of  their  direction-cosines. 
The  result  is  derived  from  (5)  by  the  form 
sm^v  =  1  —  cos'^v  =  cos^a  -\-  cos'^/?  +  cos^;/  —  cos^t'. 
To  simplify  the  writing  we  shall  omit  the  letters  cos,  using 
a,  /3,  y  and  «',  y5',  y'iov  the  direction-cosines  of  the  respec- 
tive lines.     Then 
sXn^v  =  a^ -\-  fi' -\-  y^  —  Qo^'^v 

=  a'^  /5^-f  y'-  a' a"-  /3'/3"-  y'y'' 

-  2aa'/3/3'  -  2P/3'vy'-  2yy'aa' 
=  «»(1  _  ^-)  4.  j3\l  -  ^-)+  y\l  -  y^) 

-2(af«'/?y^'  +  etc.) 
=  a\r-^r  yn  +  /5^(k"+  oc'^)  +  y\oc'^-\-  D 

-2{aa'/3/3'  -^etc.) 
=  a'/^"-{-  a''j3'-  2aa'p/3'-{-  /5V'^+  /3'Y-  ^^^'yy' 

+  y^a'^-^  y"a'-  2yy'aa' 
=  {ajS'-  a'PY-{-  {Pf-  fiyy-\-  {ya'-  y'a)\        (10) 
which  is  the  required  expression. 

Cor.     If  tivo   lines  have  the  direction- cosines  of  the  one 
respectively  equal  to  the  corresponding  ones  of  the  other,  the 
lines  are  parallel. 
For,  if 

a  =  a',         ^  =  /?',  y  =  f, 

the  last  equation  reduces  to 

sin  V  =  0. 


236  GEOMETRY  OF  THREE  DIMENSIONS. 


EXERCISES. 

1.  If  a  line  make  equal  angles  with  the  co-ordinate  axes, 
what  are  these  angles,  and  what  angle  does  it  form  with  the 
co-ordinate  planes?  Aiis.  54°  44'.  1;  35°  15'. 9. 

2.  Find  the  direction-cosines  of  a  line  which  makes  equal 
angles  with  the  axes  of  X  and  Y,  but  double  the  common 
value  of  those  equal  angles  with  the  axis  of  Z,  and  show  that 
the  angles  may  be  either  90°,  90°,  180°;  45°,  45°,  90°;  or  135°, 
135°,  270°. 

3.  In  a  room  15  by  20  feet  and  10  feet  high,  a  line  is 
stretched  from  the  northwest  corner  of  the  ceiling  to  the 
southeast  corner  of  the  floor.  Find  its  length,  the  angles 
which  it  forms  with  the  three  bounding  edges  of  the  walls 
and  ceiling,  and  with  the  walls  and  ceiling. 

4.  What  angles  do  lines  having  the  following  direction- 
vectors  form  with  the  co-ordinate  axes? — 


Line  A, 

I  =  1; 

m  =  2; 

71  =  2; 

Line  B, 

I  =  3; 

m  =  2; 

71  =  1; 

Line  0, 

I  =  2; 

m  =  3; 

71  =  4.; 

Line  D, 

i^p; 

m  =  2p; 

71  =  Sjp. 

5.  If  the  direction-cosines  of  a  line  are  proportional  to  the 
fractions  ^,  -J,  ^,  what  are  the  smallest  integers  which  we  can 
employ  as  direction- vectors? 

6.  Find  the  values  of  the  direction-cosines  of  a  line  which 
satisfy  the  equation 

cos  a  =  2  cos  J3  =  3  cos  y, 

and  the  least  integers  which  can  be  used  as  direction-vectors. 

7.  Find  the  direction-cosines  of  lines  Joining  the  following 
pairs  of  points: 

(a)  From  tlie  origin  to  the  point  (2,  3,  4); 

(b)  From  the  origin  to  the  point  (—  2,  —  3,  —  4); 

(c)  From  the  point  (1,  1,  1)  to  the  point  (2,  3,  —  1); 

(d)  From  the  point  (1,  2,-  3)  to  the  point  (-1-3,  3). 

If  the  order  of  the  points  of  each  pair  be  reversed,  what  effect 
will  this  change  have  on  the  direction-cosines? 


POSITION  AND  DIRECTION  IN  SPACE. 


237 


8.  What  angle  is  formed  by  the  two  lines  passing  from 
the  origin  to  the  points  (1,  1,  2)  and  (2,  3,  4)  respectively? 

9.  Find  the  angle  whose  vertex  is  at  the  point  (2,  3,  4), 
and  whose  sides  pass  through  the  points  (1,  2,  3)  and  (3,  5,  5). 

10.  What  angle  is  contained  by  two  lines  whose  direction- 
vectors  are  : 

Line  A,     /  =  -|-l;     m  =  -\-  3;     n  =  —  5; 
Line  B,     I  =  -  3;     m  =  +  2;     n  =  -\- 1. 

219.   Transformation  of  Co-ordinates. 

Case  I.  Transformation  to  a  new  system  whose  axes  are 
parallel  to  those  of  the  first  system.  Let  the  co-ordinates  of  a 
point  P  referred  to  the  old  system  be  x,  y  and  z,  and  let  the 
co-ordinates  of  the  new  origin  referred  to  the  old  system  be 
a,  ^  and  c.  It  is  now  required  to  express  the  co-ordinates 
x',  y',  z'  of  P  referred  to  the  new  system. 

Because  the  new  and  old  co-ordinate  planes  are  parallel, 
the  perpendiculars  dropped  from  the  point  P  upon  corre- 
sponding planes  will  be  coincident,  and  that  portion  of  a 
perpendicular  intercepted  between  the  parallel  planes  will  be 
a,  b  or  c  according  as  the  plane  is  YZ,  ZX  or  XY.  The 
difference  between  the  co-ordinates  will  therefore  be  equal  to 
these  same  quantities,  and  we  shall  have 


X'  :=^x  -a\        y  =y   -^, 
or  x   =x'  ^a-,         y   =  /  +  &; 

Z 
220.  Case  II.     Transforma- 
tion to  a  neiu  rectangular  system 
having  the  same  origin  hut  differ- 
ent directions.     Let   0-XYZ  be 
the  axes  of  the  old  system,  OX' 
any  axis  of  the  new  system,  and 
P  a    point    whose    co-ordinates 
are  to  be  expressed  in  both  sys- 
tems. ( 
From  P  drop  PQA.  plane  XY, 
From  Q  drop  QRL  axis  OX. 


(11) 


axis  OZ. 


238  QEOMETBY  OF  THREE  DIMENSIONS. 

Then,  calling  x,  y  and  z  the  co-ordinates  of  P  referred  to 
the  old  system,  we  shall  have,  by  §  213, 

X  =  OR', 

y  =  RQ\ 

z  =  QP. 

From  the  points  P,  Q  and  R  drop  perpendiculars  upon  the 
new  axis  OX',  meeting  it  in  the  points  P',  Q'  and  R' .  Let 
us  then  put 

or,  /?,  /,  the  angles  which  OX' ,  the  axis  of  the  new  sys- 
tem, makes  with  the  respective  axes  of  X,  Y  and  Z  in  the 
old  system.     We  shall  then  have 

OR'   =  OR  cos  XOX'  =  X  cos  a\ 

also,  because  RQ  \\  OY, 

R'Q'  =  RQ  cos  YOX'  =  y  cos  /3; 
also,  because  QP  \\  OZ, 

Q'P'  =  QP  cos  ZOX'  =  z  cos  y. 

Now,  the  line  OP'  is  the  algebraical  sum  of  the  three  seg- 
ments OR',  +  R'Q'y  +  Q'P',  each  segment  being  taken  posi- 
tively or  negatively  according  as  the  angle  a,  /3  ov  y  is  acute 
or  obtuse. 

Hence 

OP'  =  X  cos  a  -\-  y  COB  p  -\-  z  cos  y.  (12) 

If  we  suppose  OX'  to  represent  the  new  axis  of  X,  then 
OP'  will  be  the  co-ordinate  x  referred  to  the  new  axis,  which 
we  call  x'.  In  the  same  way  we  have  y'  and  z'  when  OX* 
represents  the  corresponding  axes.     If,  therefore,  we  put 

(X',X),(X',  Y),{X',Z)  the  angles  made  byX  with  X,  F&  Z; 
(F',X),(V',F),  (T',^)  the  angles  made  by  F' wij:h  X,  F&  Z; 
\Z',X),  (Z',  Y),  (Z',Z)  the  angles  made  by  Z'  with  X,  Y&  Z, 

we  shall  have 

x'  =  x  cos  (X',X)  +  ^cos  (X',  Y)  +  z  cos  (X',Z); ) 
?/'=a;cos(r',X)  +  ?/cos(F',r)4-^  cos  (F',Z);  [  («) 
z'  =  x  cos  (^',X)  -h  2/  cos  (Z',  Y)-\-z   cos  (Z',  Z).  ) 


x' 

= 

X  COS  a 

y' 

=: 

X  cos 

a' 

z' 

= 

X  COS 

a" 

POSITION  AND  DIRECTION  IN  SPACE.  239 

The  reliition  of  the  symbols  {X',X)y  etc.,  to  the  symbols 
X,  y,  z  and  a:',  ?/',  2;',  which  is  readily  seen,  renders  these  equa- 
tions easy  to  write.  But  the  subsequent  management  of  the 
equations  will  be  more  simple  if  we  retain  the  symbols  a,  j3 
and  y,  putting 

a,  ^  and  y  for  {X',X),{X' ,Y)  and  (X',Z); 
a',  f3'  and/  for  (r',.r),(F',  F)  and  (F',Z); 
a",  ft"  and  y"  for   {Z',X),{Z\  Y)  and  (Z',  Z). 

The  equations  (a)  will  then  be  written 

-f  y  COS  y5     -{-5;  cos  ;/;     j 

-i-y  cos  /3'    -\-z  cos  ;r';    [■         (13) 

+  2^  cos  /5"  4-  2^  cos  y''.   ) 

Each  set  of  these  cosines  must  separately  satisfy  the  equa- 
tion (7),  which  gives  the  first  three  equations  written  below. 
The  last  three  are  obtained  by  the  consideration  that,  by  §  216, 
the  cosine  of  the  angle  between  the  axes  of  X'  and  Y'  is 

cos  a  cos  a'  -\-  cos  ft  cos  ft'  -j-  cos  y  cos  y'. 

But,  because  the  new  axes  are  rectangular,  this  cosine  must 
be  zero,  as  must  also  be  the  cosines  of  the  angles  between  Y' 
and  Z',  and  between  Z'  and  X\  Thus  we  have  the  six  equa- 
tions of  condition, 

cos^<^     +  cos^/?     +  cosV     =  1; 

cos^a'    +  cos^/5'    +  cos''/'    =  1; 

cosV  +  co^'ft"  +  cosV"  =  1; 

cos  a    cos  a'  -j-  cos  ft    cos  ft'  -\-  cos  y    cos  y'  =  0; 

cos  «'  cos  a"  +  cos /5'  cos /5"-|- cos  ;k'  cos  ;/"=  0; 

cos  a"  cos  a  -|-  cos  /5"  cos  ft   +  cos  /"  cos  y   =  0. 

There  being  six  separate  equations  of  condition  between 
the  nine  cosines,  it  follows  that  all  nine  of  them  can  be  ex- 
pressed in  terms  of  some  three  independent  quantities.  How 
this  can  be  done  we  shall  show  hereafter. 

321.  We  next  remark  that  we  can  express  the  co-ordi- 
nates X,  y  and  z  in  terms  of  x',  y'  and  z' ,  by  reasoning  exactly 
as  we  have  reasoned  in  the  reverse  case,  thus  obtaining 


(14) 


g40  GEOMETRY  OF  THREE  DIMENSIONS. 

X  =  x'  cos  cc  -\-  ff  cos  «'  +  z'  cos  a";   \ 

?/  =  .t'cos/?  +  ?/'cos/?'+2'cos/5";   V  (15) 

z  =  x'  cosy  -\-  if  cos  y'  +  2;'  cos  y".    ) 

We  can  also  derive  the  first  of  these  equations  directly 
from  (12)  by  multij)lying  the  first  by  cos  a,  the  second  by 
cos  a'  and  the  third  by  cos  a",  and  adding,  noting  the  ap- 
plication of  the  results  of  §  216  to  the  angles  formed  by  the 
axes. 

Continuing  the  reasoning,  we  are  led  to  the  six  equations 
of  condition, 


cos^a:  +  cosV  +  cosV  =  1; 

cos'/5  +  cos^/J'  +  cos^/J"  =  1; 

cos";/  +  cosY'  +  cos'^;/"  =  1; 

cos  a  cos  /?  4-  cos  a'  cos  /?'  +  cos  a"  cos  /3"  =  0; 

cos  /3  cosy  ^  cos  /5'  cos  y'  +  cos  /5"  cos  y"  =  0; 

cos  y  cos  a  -{-  cos  7'  cos  a'  +  cos  ;k"  cos  a"  =  0. 


(16) 


In  reality  these  equations  are  equivalent  to  the  equations 
(14),  and  the  one  set  can  be  deduced  from  the  other  by  alge- 
braic reasoning,  without  any  reference  to  co-ordinates. 

222.  Polar  Co-ordinates  in  Space.  In  space,  as  in  a 
plane,  the  position  of  a  point  is  determined  when  its  dirediori 
and  distance  from  the  origin  are  given. 

In  space  the  direction  requires  two  data  to  determine  it. 
These  data  may  be  expressed  in  various  ways,  of  which  the  fol- 
lowing is  the  most  common.  We  take,  for  positions  of  refer- 
ence: 

1.  A  fixed  plane,  called  the /?m^amew^aZjt??awe.  For  this 
the  plane  of  XY  in  rectangular  co-ordinates  is  generally 
chosen. 

2.  An  origin  or  pole,  0,  in  this  plane. 

3.  A  line  of  reference,  for  which  we  commonly  choose  the 
axis  of  X. 

Let  P  be  the  point  whose  position  is  to  be  defined.  We 
first  have  to  define  the  direction  of  the  line  OP.  From  any 
point  P  of  this  line  drop  a  perpendicular  PQ  upon  the  fun- 


POSITION  AND  DIRECTION  IN  SPACE. 


241 


damcntal  plane,  and  join  0(2.     The  direction  of  OP  is  then 

defined  by  the  following  two  r^ 

angles: 

(1)  The  angle  P0(>  which 
OP  forms  with  its  projection 
OQ]  that  is,  the  angle  be- 
tween OP  and  the  plane. 

(2)  The  angle  X0§  which 
the  projection  of  OP  makes 
with  OX. 

It  will  be  remarked  that 
the   planes  of  these   two  angles  are  perpendicular  to  each 
other. 

To  show  that  these  two  angles  completely  fix  the  direction 
of  OP,  we  first  remark  that  when  the  angle  XOQ  is  given, 
the  line  OQ  \s  fixed. 

Next,  because  PQ  is  perpendicular  to  the  plane,  the  point 
P  and  therefore  the  line  OP  must  lie  in  the  plane  ZOQ, 
which  is  fixed  because  its  two  lines  O^and  OQ  are  fixed.  If 
the  angle  QOP  in  this  (vertical)  plane  is  given,  there  is  only 
one  line,  OP,  which  can  form  this  angle. 

Hence  the  direction  of  the  line  OP  is  completely  deter- 
mined by  the  two  angles  XOQ  and  QOP',  and  when  the  dis- 
tance OP  is  given,  the  point  P  is  completely  fixed. 

We  use  the  notation 

cp,  the  angle  QOP,  or  the  elevation  of  OP  above  the 
plane  XOY.     We  may  call  this  angle  the  latitude  of  P. 

A,  the  angle  XOQ  which  OQ,  the  projection  of  OP, 
makes  with  OX.     We  may  call  this  angle  the  longitude  of  P, 

r,  the  length  of  OP. 

Because  the  quantities  cp,  \  and  r  completely  fix  the  posi- 
tion of  P,  they  are  called  the  polar  co-ordinates  of  P  in 
space. 

233.  Relation  of  the  Preceding  System  to  Latitude  and 
Longitude.  For  another  conception  of  the  angles  q)  and  A, 
pass  a  sphere  around  0  as  a  centre,  and  mark  on  its  surface 
the  points  and  lines  in  which  the  lines  and  planes  belonging 
to  the  preceding  figure  intersect  it.     Then 


242 


GEOMETRY  OF  THREE  DIMENSIONS. 


The  fundamental  plane  OXQ  intersects  the  spherical  sur- 
face in  the  great  circle  XQY\ 

The  line  OX  intersects  it  in  X; 

The  line  OQ  intersects  it  in  Q-, 

The  lines  OP  and  OZ  intersect  it  in  P  and  Z. 

We  therefore  have 

Angle  XOQ  measured  by  arc  XQ\ 
Angle  QOP  measured  by  arc  QP, 


If  now  we  imagine  this  sphere  to  be  the  earth,  the  great 
circle  XP"  to  be  its  equator,  Z  to  be  one  of  the  poles,  and  P 
any  point  on  its  surface,  then 

The  arc  QP  or  the  angle  QOP  is  the  latitude  of  P; 

The  arc  XQ  or  angle  XOQ  is  the  longitude  of  P,  counted 
from  ZX  as  a  prime  meridian.  Thus  the  angles  we  have 
been  defining  may  be  described  under  the  familiar  forms  of 
longitude  and  latitude. 

224,  Peoblem.  To  transform  the  position  of  a  point 
from  rectangular  to  polar  co-ordinates,  and  vice  versa. 

Comparing  the  definitions  of  rectangular  and  polar  co- 
ordinates, we  put,  for  the  point  P  in  §  222, 

PQz=z=:  OP  sin  ^; 
OQ  =  OP  cos  (p. 


POSITION  AND   DIRECTION  IN  SPACE.  243 

Now,  supposing  a  perpendicular  dropped  from  Q  upon  OX, 
this  perpendicular  will  be  the  ordinate  y,  and  will  meet  OX 
at  the  distance  x  from  the  origin.     Thus, 

x=  OQ  cos  XOQ  =  OQ  cos  ?i; 
y=OQ  sin  XOQ  =  0^  sin  X. 

Putting  OP  =  r,  and  substituting  for  0§  its  value,  we  have 

ic  =  r  cos  cp  cos  A;   j 

y  =  r  cos  99  sin  A;   >•  (17) 

z  =  r  sm  cp'j  ) 

which  are  the  required  equations. 

Cor.  The  direction-cosines  of  the  line  OP  in  terms  of  cp 
and  A  are 

cos  a  =  cos  q)  cos  A;   j 

cos  /?  =  cos  cp  sin  A;   >•  (18) 

cos  y  =  sin  cp.  ) 

235,  The  result  stated  in  §  220,  that  the  nine  direction- 
cosines  of  one  system  of  rectangular  axes  with  respect  to 
another  system  can  be  expressed  in  terms  of  three  independ- 
ent quantities,  may  now  be  proved  as  follows: 

1.  Let  OP  be  any  one  accented  axis,  say  JT';  the  direction- 
cosines  of  this  axis  are  expressed  in  terms  of  two  angles, 
(p  and  A,  by  (18). 

2.  Imagine  a  plane  =31,  passing  through  0,  §223,  per- 
pendicular to  OP.  This  plane  M  will  be  completely  deter- 
mined by  the  direction  OP;  whence  the  line  =  iV'in  which  it 
cuts  the  fundamental  plane  XY  will  also  be  determined. 

3.  The  new  axis  V  may  lie  in  any  direction  from  the 
point  0  in  the  plane  M.  One  more  angle  =  ?/?  is  required  to 
determine  this  direction,  and  for  this  angle  we  may  take  the 
angle  which  F'  forms  with  the  line  iV. 

4.  The  direction  of  the  axis  Z'  is  then  completely  fixed, 
because  it  must  lie  in  the  plane  31  and  make  an  angle  of  90° 
with  Y\ 

Thus,  cp,  A  and  tp  completely  determine  the  directions  of 
the  three  new  axes. 


244  GEOMETRY  OF  TUREE  DIMENSIONS. 

EXERCISES. 

1.  If,  in  the  figure  of  §  223,  the  co-ordinates  of  the  point 

P  are 

a;  =  27,         y  =  l^,         ^  =  17, 

find  its  polar  co-ordinates  r,  q)  and  \. 

Method  of  Solution.     The  quotient  of  the  first  two  equations  (17)  gives 

tan  A  =  ^, 

X 

from  which  A.  is  found.     Then  we  find  sin  X  or  cos  X  or  both,  and  com- 
pute 

X  y 

r  cos  cp  =  r-  =  - — Y' 

cos  X       sin  X 

Next  we  have 

z 

tan  cp  = , 

r  cos  q) 

from  which  we  find  <p.     Then 

z      _  r  cos  (p 
""  siu  ^  "    cos  cp  ' 

2.  Supposing  the  radius  of  the  earth  to  be  6369  kilometres, 
the  longitude  of  New  York  to  be  74°  west  of  Greenwich,  and 
its  latitude  to  be  40°  32',  it  is  required  to  find  the  rectangular 
co-ordinates  of  New  York  referred  to  the  following  system  of 
axes  having  the  earth's  centre  as  the  origin: 

Xin  the  equator,  and  on  the  meridian  of  Greenwich. 
Y  in  the  equator,  in  longitude  90°  east  of  Greenwich . 
Z  passing  through  the  North  Pole. 

3.  If,  in  the  figure  of  §  222,  we  take  a  point  P'  whose 
latitude  is  the  same  as  that  of  P  and  whose  longitude  is  90° 
greater  than  that  of  P,  it  is  required  to  express  the  angle 
POP\ 

4.  If  the  angle  <p  is  negative,  within  what  region  will  the 
point  P  be  situated? 

5.  If  we  take  a  point  P'  whose  latitude  is  cp  and  whose 
longitude  is  A  +  180°,  how  will  it  be  situated  relatively  to  P, 
and  what  will  be  the  angle  POP'? 

6.  If  we  take  a  point  P'  for  which 

(p'  =  180°  -  cp, 

r  =x-^  180°, 

show  that  this  point  will  be  identical  with  P. 


CHAPTER      II 

THE    PLANE. 


226.  Introductory  Considerations  on  the  Loci  of  Equa- 
tions. If  the  values  of  the  three  co-ordinates  of  a  point  are 
not  subject  to  any  restriction,  the  point  may  occupy  any  posi- 
tion in  space.  Kestrictions  upon  the  position  are  algebraically 
expressed  by  equations  of  condition  between  the  co-ordinates. 
Let  us  inquire  Avhat  will  be  the  locus  of  the  point  when  the 
co-ordinates  are  required  to  satisfy  a  single  equation  of  condi- 
tion. By  means  of  such  an  equation  we  may  express  any  one 
of  the  co-ordinates,  z  for  example,  in  terms  of  the  other  two, 
the  form  being 

We  can  now  assign  any  values  we  please  to  x  and  y,  and 
for  each  pair  of  such  values  find  the  corresponding  value  of  z. 
To  each  pair  of  values  of  x  and  y  will  correspond  a  certain 
point  on  the  plane  of  JTY.  If  at  this  point  we  erect  a  per- 
pendicular equal  to  z,  the  end  of  each  perpendicular  will  be  a 
point  whose  co-ordinates  satisfy  the  equation. 

We  may  conceive  these  perpendiculars  to  become  indefi- 
nitely numerous  and  indefinitely  near  each  other,  thus  tending 
to  form  a  solid.  Their  ends  will  then  tend  to  form  the  sur- 
face of  this  solid.  But  these  ends  are  the  locus  of  the  equation 
(a).     Hence 

The  locus  of  a  single  equation  of  condition  among  the  co- 
ordinates is  a  surface. 

If  a  second  equation  is  required  to  subsist  among  the  co- 
ordinates, the  locus  of  this  equation  will  be  a  second  surface. 

If  the  co-ordinates  are  required  to  fulfil  both  conditions 
si7nultaneously J  then  the  point  must  lie  in  both  surfaces;  that 


246  GEOMETRY  OF  THREE  DIMENSIONS. 

is,  it  must  lie  on  the  line  in  which  the  surfaces  intersect. 
Hence 

The  locus  of  two  simultaneous  equations  betioeen  the  co- 
ordinates is  a  line, 

2^7.  To  find  the  Eq^iation  of  a  Plane.  The  property  of 
a  plane  from  which  the  locus  can  be  most  elegantly  deduced  is 
this:  If  on  any  line  which  intersects  the  plane  perpendicularly 
we  take  two  points,  A  and  B,  equidistant  from  the  point  of 
intersection  and  on  opposite  sides,  then  every  point  of  the 
plane  will  be  equidistant  from  A  and  By  and  no  point  not  on 
the  plane  will  be  equidistant. 

Let  us  then  drop  from  the  origin  a  perpendicular  upon  the 
plane,  and  continue  it  to  a  distance  on  the  other  side  equal  to 
its  length,  and  let  P  be  the  point  at  which  it  terminates. 
The  condition  that  a  point  shall  lie  on  the  plane  will  then  be 
that  it  shall  be  equidistant  from  the  origin  and  from  P, 

Let  us  put 

a,  h,  c,  the  co-ordinates  of  P ; 

X,  y,  z,  the  co-ordinates  of  any  point  on  the  plane. 

Then,  by  §§  214,  215,  the  squares  of  the  distances  of  (x,  y,  z) 
from  the  origin  and  from  P  will  be  respectively 

x^^y'-^rz^ 
and  {x  -  ay  -\- {y  -  hy  +  (^  -  cy. 

Developing  and  equating  these  two  expressions,  we  find,  for 
the  required  equation, 

2ax  +  Uy  +  2cz  =  a'  +  b'  -\-  c\ 

To  reduce  this  equation,  let  us  put 

Py  the  length  of  the  perpendicular  dropped  from  the  origin 
upon  the  plane;  that  is,  one  half  the  line  from  the  origin  to  P  ; 
a,  /3,  y,  the  angles  which  this  perpendicular  makes  with 
the  respective  axes  of  JT,  Y  and  Z. 

We  shall  then  have  OP  =  2p,  and  the  values  of  a,  I?  and 
c  will  be,  by  §  214, 

a  =  2p  cos  a; 
b  =  2p  cos  /?; 
c  =  2p  cos  y. 


THE  PLANE.  247 

Substituting  these  values  in  the  equation  of  the  plane,  reduc- 
ing, and  remarking  that 

cos''^  +  cos'/?  +  cos';/  =  1, 
the  equation  of  the  plane  becomes 

X  cos  a  -{-  y  co^  ^  -\-  z  0,0^  y  —  p  —  0,  ( 1 ) 

which  is  called  the  normal  equation  of  the  plane. 

228.  The  A^igles  a,  fi  and  y.  As  we  have  defined  the 
angles  «,  /?  and  y,  they  are  the  angles  which  the  perpen- 
dicular p  forms  with  the  co-ordinate  axes.  It  is  shown  in 
Solid  Geometry  that  the  angle  between  any  two  planes  is  equal 
to  that  between  any  two  lines  perpendicular  to  them.    Because 

Plane  YZ  ±  axis  X, 

Plane  (1)   ±  line  p, 

.  • .  plane  (1)  makes  the  angle  a  with  the  plane  YZ.     Hence 

we  may  define  a,  ft  and  y  as  the  angles  which  the  plane  mahes 

with  the  co-ordinate  planes  YZ,  ZX  and  XY  respectively. 

One  restriction  upon  this  proposition  is  necessary.  Two 
supplementary  angles  are  formed  by  any  two  planes,  so  that, 
in  the  absence  of  any  convention,  we  should  say  that  the 
angle  between  the  planes  is  either  equal  or  supplementary  to 
that  between  the  lines.  The  best  way  of  avoiding  ambiguity 
is  to  choose,  for  the  angle  between  the  planes,  the  angle  be- 
tween the  perpendiculars  dropped  from  the  origin  upon  the 
planes. 

If  the  angles  a,  f3  and  y  are  the  same  for  several  planes, 
these  planes,  being  perpendicular  to  the  same  line,  are  paral- 
lel. Hence  they  may,  in  a  certain  sense,  be  said  to  have  the 
same  direction.  We  may  therefore  call  cos  a,  cos  y^  and  cos  y 
the  direction-cosines  of  the  plane.  They  are  also  the  direc- 
tion-cosines of  the  perpendicular  dropped  from  the  origin 
upon  the  plane. 

229.  Theorem  I.  Every  equation  of  the  first  degree  be- 
iween  rectangular  co-ordinates  in  space  is  the  equation  of 
some  plane. 

Proof     Let  the  equation  be 

Lx^My^Nz-\-D  =  0. 


248  GEOMETRY  OF  THREE  DIMENSIONS. 


Divide  this  equation  by  VU  -{-  M'^  -\-  iV%   and  determine 
three  angles,  a,  /?  and  y,  by  the  equations 

L 


cos  a 


COS  p 


(2) 


COS  y  —  —  —  . 

This  will  always  be  possible,  because  each  of  these  cosines 
is  less  than  unity.     Because  these  cosines  fulfil  the  condition 

(7),  B  217,  we  can  draw  a  line  of  length  =     , 

from  the  origin  making  the  angles  a,  §  and  y  with  the  sev- 
eral co-ordinate  axes.  Through  the  end  of  this  line  pass  a 
plane  perpendicular  to  it.  Then,  by  the  last  section,  the  equa- 
tion of  this  plane  will  be 

a;  cos  a  -I-  V  cos  /?  +  2?  cos  1/  -] 3=  —  0, 

an  equation  which  becomes  identical  with  that  assumed  in 
the  hypothesis  by  clearing  of  denominators  and  substituting 
from  (2).  We  may  therefore  put  the  theorem  in  the  following 
more  specific  form: 

Every  equation  of  the  form 

Lx^  My  -^  Nz-\-  D  =  0 

represents  a  certain  defitiite  plane,  namely y  the  plane  passing 
perpendicularly  through  the  end  of  that  line  tuhich 

emanates  from  the  origin; 

makes  with  the  axes  angles  whose  cosines  are 

L  M  ^  N 

—  and 


respectively; 

and  has  the  lenqth  —t=~ — -^— 

VU  +  M'  +  N' 


THE  PLANE.  249 

230.  Notation,     By  "  the  plane  (L,  M,  N,  D)"  we  mean 
''  the  plane  whose  equation  is 

Lx  +  Ml/  -{-  JVz-{-I)  =  0." 

Vef.  An  equation  of  a  plane  in  which  the  four  quantities 
Z,  if,  a  and  D  are  all  independent  is  called  the  general 
equation  of  a  plane. 

The  general  equation  may  be  considered  as  related  in  two 
ways  to  the  normal  or  other  special  forms  of  the  equation. 

I.  The  special  forms  are  cases  in  which  certain  relations 
exist  among  the  quantities  L,  M,  iVandi).  For  example,  the 
normal  form  is  the  special  case  in  which  U  -\-  M^  -{-N"^  =  1. 

Whenever  we  find  this  condition  satisfied,  we  know  that 
the  equation  is  in  the  normal  form. 

II.  The  general  equation  may  always  be  reduced  to  the 
normal  form  by  dividing  by   V U  -f  if ^  -J-  N"^. 

Direction-  Vectors.  From  the  equation  (2)  it  is  seen  that 
L,  M  and  N  may  be  taken  as  the  direction-vectors  of  any  line 
perpendicular  to  the  plane,  because  they  are  severally  equal  to 
the  direction-cosines  of  such  a  line  multiplied  by  the  common 
factor  \/{U  -+-  M"^  +  N^)-     Hence  we  conclude: 

The  equation  of  every  plane  perpendicular  to  a  line  whose 
direction-vectors  are  I,  m  and  n  may  he  luritten  in  the  form 

lx  -f  my  -\-  nz  -\-  d  =  0', 

and,  conversely,  /or  the  direction-vectors  of  any  line  perpen- 
dicular to  the  plane  {L,  M,  N,  D)  may  he  taTcen  L,  M  and  N. 

231.  Special  Positions  of  a  Pla7ie.  If  one  of  the  co- 
efficients L,  M,  or  ^Y  vanishes,  the  cosine  of  the  angle  which 
the  plane  makes  with  the  corresponding  co-ordinate  plane  will 
also  vanish;  that  is,  the  plane  will  be  i^erpendicular  to  the  co- 
ordinate plane,  and  therefore  parallel  to  the  axis  of  that  plane. 
Hence  a7i  equation  of  the  first  deyree  hetioeen  two  only  of  the 
co-ordinates  represents  a  2)lane  parallel  to  the  axis  of  the  miss- 
ing co-ordinate. 

For  example,  the  locus  of 

Lx-\-My^D  =  0 
is  a  plane  parallel  to  the  axis  of  Z. 


250  GEOMETRY  OF  TUREE  DIMENSIONS. 

It  follows  that  if  two  co-ordinates  are  missing,  the  locus 
will  be  parallel  to  the  common  plane  of  the  missing  co-ordi- 
nates.    For  example,  the  locus  of 

Lx  -^  D  =  0 
D 

will  be  a  plane  perpendicular  to  the  axis  of  X  and  parallel  to 
the  plane  YZ. 

233.  Lines  and  Points  coujieded  with  a  Plane.  The 
following  lines  and  points  are  determined  by  every  plane: 

I.  The  three  lines  in  which  it  intersects  the  co-ordinate 
planes. 

II.  The  three  points  in  which  it  intersects  the  co-ordinate 
axes. 

III.  The  foot  of  the  perpendicular  from  the  origin  upon 
the  plane. 

When  we  include  among  possible  lines  and  points  the  lines 
and  points  at  infinity,  the  above  three  lines  and  four  points 
will  always  be  determinate. 

Def.  The  lines  in  which  a  plane  intersects  the  co-ordinate 
planes  are  called  traces  of  the  plane. 

The  distances  from  the  origin  to  the  three  points  in  which 
a  plane  cuts  the  co-ordinate  axes  are  called  the  intercepts 
of  the  axes  by  the  plane. 

233.  Problem.  To  find  the  equations  of  the  traces  of  a 
plane. 

The  trace  of  the  plane  upon  the  plane  of  YZ  is  simply 
those  points  of  the  plane  for  which  ic  =  0.  Hence,  if  we  put 
X  =  0  in  the  equation  of  a  plane,  we  have  the  equation  of  its 
trace  upon  YZ.     Therefore,  in  the  general  equation 

Lx-^  My  -\-  Nz-{-  D=  0, 

the  equations  of  the  traces  upon  the  co-ordinate  planes  are: 

On  YZ,  My  +  Xz  ^  D  =  0-, 
On  ZX,  Lx  -\-  Nz  -\-  D  =  (); 
On  XY,     Lx  +  My+D^  0. 


THE  PLANE. 


251 


These  equations,  representing  lines  upon  planes,  can  be  dis- 
cussed like  the  equations  of  lines  in  Plane  Analytic  Geometry. 

234.  Problem.  To  express  the  lengths  of  the  intercepts 
of  the  axes  hy  a  plane. 

At  the  point  where  the  plane  cuts  the  axis  of  Xwe  have 
y  =  0  and  2;  =  0.  Hence  the  intercept  is  the  value  of  x  cor- 
responding to  zero  values  of  y  and  z,  and  so  with  the  other 
co-ordinates.     Thus: 


Intercept  on  X  = 
Intercept  on  Y  = 
Intercept  on  Z  = 


n 

D 

D 

N' 


(4) 


Scholium.  Each  of  the  traces  necessarily  meets  the  other 
two  on  the  several  co-ordinate  axes,  and  their  points  of  meet- 
ing are  those  in  which  the  plane  cuts  the  axes.  Hence  the 
traces  form  a  plane  triangle  of  which  the  points  in  which  the 
plane  intercepts  the  axes  are  the  vertices. 

Each  of  the  sides  of  this  triangle  is  the  hypothenuse  of  a 
right  triangle  of  which  the  sides  containing  the  right  angle 
are  the  intercepts  upon  the  axes. 

The  relations  between  the  sides  and  angles  of  these  tri- 
angles, considered  individually,  may  be  investigated  by  the 
methods  of  Plane  Trigonometry. 

235.  Problem.  To  express  the  equation  of  a  plane  171 
terms  of  its  intercepts  upon  the  axes 

Let  us  put 

a,  I,  Cy  the  intercepts  on  the  axes  of  X,  Y  and  Z  respec- 
tively. 


Then 


a  =  — 


D 

■  -'-',■. 

D 

.. .,.--.. 

D 

c 

(5) 


252  GEOMETRY  OF  THREE  DIMENSIONS. 

Substituting  these  values  of  L,  M  and  N  in  the  general  equa- 
tion, it  reduces  to 

1+1+^  =  ^'  («) 

which  is  the  required  equation. 

EXERCISES. 

1.  Write  the  equation  of  that  plane  for  which  the  co- 
ordinates of  the  foot  of  the  perpendicular  from  the  origin 
upon  the  plane  are  1,  2,  and  3. 

2.  If,  in  the  general  equation  of  a  plane,  the  coefficients 
L,  Jtf  and  iV^are  all  equal,  what  angle  will  the  perpendicular 
make  with  the  co-ordinate  axes,  and  Avhat  angle  will  the 
plane  make  with  the  co-ordinate  planes? 

3.  The  equation  of  a  plane  being 

3a;  +  4?/  -  122;  =  26, 

it  is  required  to  reduce  it  to  the  normal  form  to  find  the 
angles  which  it  forms  with  the  co-ordinate  axes,  the  equations 
of  its  traces  upon  the  co-ordinate  planes,  the  lengths  of  its  in- 
tercepts upon  the  co-ordinate  axes,  the  lengths  of  its  traces 
between  these  intercepts,  and  its  least  distance  from  the 
origin. 

4.  The  intercepts  being  in  the  proportion  1:2:3,  what 
are  the  cosines  of  the  angles  which  the  perpendicular  upon 
the  plane  makes  with  the  axes? 

5.  Show  that  the  inverse  square  of  the  perpendicular  from 
the  origin  upon  a  plane  is  equal  to  the  sum  of  the  inverse 
squares  of  its  intercepts;  i.e., 

1  =  1  +  1  +  1. 

p         a         0         c 

6.  Show  the  corresponding  relation  between  the  two  sides 
of  a  right  triangle  and  the  perpendicular  from  the  vertex 
upon  the  hypothenuse. 

7.  Express  the  lengths  of  the  sides  of  the  triangle  formed 
by  the  traces  of  a  plane  in  terms  of  the  intercepts,  and  prove 
that  the  sum  of  the  squares  of  the  sides  is  twice  the  sum  of 
the  squares  of  the  intercepts. 


THE  PLANE.  253 

8.  A  plane  cuts  traces  whose  lengths  between  the  axes  are: 
On  plane  YZ,  u\     on  plane  ZX,  v;     on  plane  XY,  w. 

Find  the  lengths  of  the  intercepts  and  the  equation  of  the 
plane  in  terms  of  s^  =  ^(u^  +  ^^  +  w^). 

Ans.  — ^=  +  -JL^  +  -=^=  =  1. 

9.  Find  the  angle  included  between  the  planes 

and  x-y-{-2z  =  b.    (Comp.  §§  21G,  230) 

236.  Plane  satisfying  Given  Conditions.  If  a  plane  is 
required  to  satisfy  a  condition,  that  condition  can  be  expressed 
as  an  equation  between  the  constants  L,  M,  N,  D,  which  de- 
termine the  position  of  the  plane.  By  means  of  this  equation 
one  of  the  constants  can  be  eliminated  from  the  equation  of 
the  plane,  and  the  condition  will  then  be  fulfilled  for  all  values 
of  the  remaining  constants. 

If  two  conditions  are  given,  two  constants  can  be  elimi- 
nated; if  three,  all  the  constants.  For,  although  the  general 
equation  of  the  plane  contains  four  constants,  it  depends  only 
on  the  three  ratios  of  any  three  of  these  constants  to  the 
fourth.  In  fact,  we  can  always  reduce  the  general  equation  to 
the  form  (6) 

a^h^  c       ^' 
which  contains  but  three  arbitrary  constants. 

237.  Problem.  To  find  the  equation  of  a  plane  passing 
through  a  given  point. 

Let  {x\  y',  z')  be  the  given  point.  In  order  that  the 
plane  {L,  M,  JV,  D)  may  pass  through  this  point,  its  constants 
must  satisfy  the  condition 

Lx'  -f  My'  4-  JV>/  +  i)  =  0,  {a) 

which  gives 

D=-  (L:c'  + J//  +  iW). 

Substituting  this   value   of  D  in  the  general  equation,  the 
latter  becomes 

Lx  +  My  -\-Nz-  {Lx'  +  My'  +  Nz')  =  0,  (7) 


254  QEOMETEY  OF  THREE  DIMENSIONS. 

or,  in  another  form, 

X(:,  -  X')  +  Miy  -  y')  +  N{z  -  z')  =  0.  (8) 

Remark  1.  In  these  equations  we  may  assign  any  values 
we  please  to  L,  M  imd  iV,  witliout  the  plane  ceasing  to  pass 
through  the  point  {x\  y',  z'),  as  is  evident  from  (8). 

Remark  2.  If  we  had  two  equations  of  the  form  (a),  we 
could  eliminate  two  of  the  constants,  say  N  and  D,  and  L 
and  if  would  still  remain.  If  we  had  three  equations,  we 
could  eliminate  three  constants,  M,  iV^and  D  for  example. 
That  is,  we  could,  by  solving  the  equations,  express  M,  iV^and  D 
in  terms  of  L.  Substituting  these  values  in  the  general  equa- 
tion, the  latter  would,  it  would  seem,  still  contain  the  constant 
L.  But,  in  reality,  L  would  enter  only  as  a  factor  of  the 
whole  equation,  and  would  therefore  divide  out.  Hence,  when 
we  eliminate  any  three  of  the  four  constants,  the  fourth  drops 
out  of  itself  without  the  introduction  of  any  further  condition. 

Relations  of  Two  or  More  Planes. 

238.  Parallel  and  Perpendicular  Planes. 

Theorem  II.     If,  in  the  equations  of  any  tiuo  planes 
Lx  -f  My   -\-  Nz   -\-  D   ^  0, 
L'x  +  M'y  +  N'z  +  i)'  z=  0, 
tlie  direction-vectors  L',  M'  and  N'  are  proportional  to  L,  M 
and  N  respectively,  the  two  planes  are  parallel. 

Proof.  The  direction-vectors  of  the  perpendiculars  from 
the  origin  upon  the  planes  being  proportional,  the  direction- 
cosines  are  equal  (§  217),  and  these  perpendiculars  are  co- 
incident (§  218).  Hence  the  planes  are  perpendicular  to  the 
same  line  and  therefore  parallel.     Q.  E.  D. 

Problem.  To  find  the  condition  that  two  plajies  given  ly 
their  equations  shall  he  perpendicular  to  each  other. 

Let  the  planes  be  (Z,  M,  N,  D)  and  {U,  M',  N',  D'). 

The  planes  will  be  perpendicular  when  the  perpendiculai  s 
from  the  origin  are  perpendicular  to  each  other.  The  con- 
dition is,  from  eq.  (9)  of  §  217, 

LL'  -f  MM'  +  NN'  =  0. 


THE  PLANE.  255 

239.  Notation.  I.  We  use  tlie  symbols  P,  P',  P",  etc., 
Q,  Q' ,  etc.  etc.,  to  signify  functions  of  the  co-ordinates  of 
the  first  degree.     For  example, 

P  =  Lx  -\-  My  +  Nz  +  D', 
P'  =  L'x  +  M'y  +  N'z  +  i)'; 
etc.  etc.  etc. 

II.  By  the  expression  "the  plane  P"  we  mean  the  plane 
whose  equation  is  P  =  0. 

240.  Theorem  III.  If  in  a  function  P  we  substitute 
for  X,  y  and  z  the  co-ordinates  x' ,  y'  and  z*  of  a  point,  P  will 
then  express  the  distance  of  that  point  from  the  plane  P  =  0 
multiplied  ly  the  factor  V L"  +  if  ^  -}-  N^. 

Proof.  Let  us  pass  through  the  point  (re',  y',  z')  a  plane 
A  parallel  to  the  plane  P.  The  equation  of  this  plane  will 
be  (§  237) 

Lx  +  My  -\-Nz-  {Lx'  +  My'  +  Nz')  =  0. 

The  term  independent  of  x,  y  and  z  is  —{Lx'-^My'-^Nz')y 
which  takes  the  place  of  D  in  the  general  equation.  Hence 
the  perpendicular  distance  of  this  plane  A  from  the  origin  is 

Lx'  4-  My'  +  Nz' 


P 


VU  -\-M''  '-{-N^ 


The  perpendicular  distance  of  the  plane  P  from  the  origin  is 
(§229) 

--D 

Because  the  point  {x',  y',  z')  is  in  the  plane  A  \\  P,  the 
distance  of  (x',  y',  z')  from  the  plane  P  is  equal  to  the  con- 
stant distance  between  the  planes,  and  hence  to  the  difference 
p  —  p'  between  the  perpendiculars.     Hence 

Distance  of  (:.',  y',  z')  from  P  =  L^±BL±M±^-, 
or 


Lx'  +  My'  -f-  Nz'  +  P  =  Distance  x  ^ U  +  if  ^  ^N\ 

Q.  E.  D. 


256  GEOMETRY  OF  THREE  DIMENSIONS. 

Cor.     If  the  equation  is  in  the  normal  form,  we  shall  have 


and  tlie  expression  P  will  then  represent  the  distance  of  the 
point  whose  co-ordinates  appear  in  it  from  the  plane  P  =  0, 


EXERCISE. 

Show  that  the  angle  e  between  the  two  planes 

Lx  +  My-]-  N'z-^D  =  0  and   L'x  +  M'y  -\-  N'z-rD'  =  0 

is  given  by  the  equation 

LL'  +  MM'  +  iVN' 

cos  s  =  --  '     

i/L'  -i-M'-\-  iV'-'  V U' -i- M'' -{- N'' 

241.  Theorem  IV.  If  P  =  0  and  P'  =  0  ie  the  equa- 
tions of  any  two  planes,  and  A  and  X'  constants,  the  equation 

XP  4-  rp'  =  0  (a) 

will  he  the  equation  of  a  third  plane  intersecting  the  other  tivo 
in  the  same  line. 

Proof.  I.  The  expression  XP  +  X'P'  is  of  the  first  degree 
in  X,  y  and  z.     Therefore  the  equation  is  that  of  some  plane. 

II.  Every  set  of  values  of  the  co-ordinates  x,  y  and  z 
which  simultaneously  satisfy  both  equations  P  =  0  and  P'  =  0 
also  satisfy  equation  {a).  The  co-ordinates  which  satisfy  both 
equations  are  those  of  their  line  of  intersection  (§  226).  There- 
fore these  co-ordinates  also  satisfy  («);  whence  the  line  lies  in 
the  plane  XP  -\-  X'P',  which  proves  the  theorem. 

Cor.  If  three  functions,  P,  P'  and  P",  are  such  that  it 
is  possible  to  find  three  constant  coefiicients,  A,  A'  and  A", 
which  lead  to  the  identity 

AP  +  A'P'  +  A"P"eO, 
the  three  planes  P,  P'  and  P"  intersect  in  the  same  line. 

Theorem  V.  If  Q  =  0  and  Q'  =  0  are  the  eqiiations  of 
two  planes  in  the  normal  form,  the  equal io7is 

Q  -f  (2'  =  0        and         Q  -  Q'  =  0 


THE  PLANE.  257 

represent  the  planes  which  Used  the  dihedral  angles  formed  hy 
the  plane  Q  and  Q' . 

Proof,  Because  the  expressions  Q  and  Q'  represent  the 
distances  of  the  point  (re,  ?/,  z)  from  the  planes  Q  and  §' 
(§  240),  it  follows  that  the  equation 

q^Q'        or         §  -  §'  =  0 

will  express  the  condition  that  the  said  point  is  equally  dis- 
tant from  the  planes  Q  and  Q' .  Hence  it  lies  upon  the  plane 
bisecting  the  angle  formed  by  Q  and  Q' ,  and  this  plane  is  the 
locus  of  the  equation  Q  —  Q'  =  0. 

The  equation  Q  -\-  Q'  =  0  is  equivalent  to  Q  =  —  Q' , 
and  asserts  that  the  point  (x,  y,  z),  if  on  the  positive  side  of 
the  one  plane,  is  on  the  negative  side  of  the  other  at  an  equal 
distance.     Therefore  it  bisects  the  adjacent  dihedral  angle. 

Cor.  In  the  case  supposed,  the  two  planes  Q  —  Q^  and 
(3  -f  C'  ^^6  perpendicular  to  each  other  because  they  are  the 
bisectors  of  adjacent  angles. 

Theorem  VI.  If  A,  A'  a7id  A"  are  constant  coefficients, 
the  equation 

AP  +  A'P'+  A"P"  =  0  {h) 

represents  a  plane  passing  through  the  common  point  of  inter- 
section of  the  planes  P,  P'  and  P" , 

Proof.  In  the  same  way  as  with  Theorem  IV,  it  is 
shown  (1)  that  the  equation  is  that  of  a  plane,  and  (2)  that 
the  co-ordinates  of  any  and  every  point  common  to  the  three 
planes  P,  P'  and  P"  satisfy  equation  {Jj).  Now,  because  any 
three  planes  have  one  point  common  (which  may  be  at  infin- 
ity), the  point  common  to  P,  P'  and  P"  lies  on  the  plane 
{h).     Q.  E.  D. 

Cor.  If  four  functions,  P,  P',  P"  and  P'",  are  such  that 
an  identity  of  the  form 

\P  _|_  VP'  -f  V'P"  +  V"P"'  =  0 

is  possible,  the  four  planes  P,  P',  P"  and  P'"  will  intersect 
in  a  point. 


258  GEOMETRY  OF  THREE  DIMENSIONS. 

242.  Bisectors  of  Dihedral  Angles.  The  foregoing 
principles  enable  us  to  prove  many  elegant  relations  among 
the  planes  which  bisect  the  dihedral  angles  of  a  solid,  or  of  a 
solid  angle.     Let 

Q  =  0,         Q'  =  0,         e"  =  0, 

be  the  equations  of  any  three  planes  in  the  normal  form. 
Since  any  three  planes  meet  in  a  point,  they  may  be  consid- 
ered as  forming  a  dihedral  angle  at  that  point. 

The  bisecting  planes  of  the  three  dihedral  angles  formed 
by  the  planes  Q,  Q'  and  §"  will  be  (§  241) 

Q     -  Q'    =  0  =  P; 
Q'    _  §"  ^  0  =  P'; 
Q''  -  Q     =  0  E  P". 

These  functions  satisfy  the  condition 

P  -^  P'  -\-  P''  =  0. 

Therefore  the  three  hisecting  planes  of  a  dihedral  angle  in- 
tersect on  a  line. 

Placing  the  centre  of  a  sphere  at  the  vertex  of  the  dihe- 
dral angle,  and  considering  the  spherical  triangle  formed  by 
the  planes  Q,  §',  §",  we  have  the  theorem: 

The  great  circles  lisecting  the  interior  angles  of  a  spherical 
tria7igle  meet  in  a 


EXERCISES. 

1.  In  order  that  the  two  planes  P^P'=  0  and  P~P'—  0 
may  be  perpendicular  to  each  other,  show  that  the  coefficients 
of  X,  y  and  ;2;  in  P  and  P'  must  satisfy  the  condition 

2.  Describe  the  relative  position  of  the  four  planes 

^  +  2/  -1-  2  =  0, 

x+    g-2z  =  0, 
X  —  2g  -\-    z  =  0, 

and  find  the  angles  which  each  makes  with  the  three  others. 


THE  PLANE.  259 

3.  Show  that  the  line  of  intersection  of  the  two  planes 

ax  -\-  hy  -\-  cz  -\-  d  —  0, 
ax  -{-hy  —  cz  -\-  d  —  0, 

is  in  the  plane  of  XY,  and  that  its  equation  in  this  plane  is 

ax  -\-  hy  -\-  d  =  0. 

4.  What  is  the  condition  that  a  plane  shall  pass  through 
the  origin? 

5.  Write  the  equation  of  a  plane  making  equal  angles  with 
the  three  co-ordinate  planes  and  cutting  off  from  the  axis  of 
JTan  intercept  a. 

6.  When  a  plane  makes  equal  angles  with  the  three  co- 
ordinate planes,  what  is  the  ratio  of  each  inteicept  which  it 
cuts  off  from  the  axes  to  the  j)erpendicular  from  the  origin 
upon  the  plane?  Ans.    Vd  :  1. 

7.  Write  the  equation  of  a  plane  which  shall  make  equal 
angles  with  the  axes  of  Xand  Z,  and  shall  be  parallel  to  the 
axis  of  Y, 

8.  What  is  the  distance  apart  of  the  parallel  planes 

X  -f  2y  +  2z  =  a; 
2x -\-  4.y  -{-  4rz  =  b? 

9.  Write  the  equation  of  the  plane  which  shall  pass 
through  the  point  (1,  2,  2)  and  be  parallel  to  the  plane 

—  x-^2y  —  z  =  0. 

10.  Write  the  equation  of  the  plane  which  shall  pass 
through  the  origin,  the  point  (1,  1,  2)  and  the  point  (2, 3, 1). 

A71S.     —  6x  -{-  dy  -{-  z  =  0. 

11.  Write  the  equation  of  the  plane  which  shall  pass 
through  the  origin  and  the  point  (1,  2,  2),  and  shall  be  per- 
pendicular to  the  plane 

X  —  y  -^  z  =  0. 

Ans.     4:X  -f  y  —  3z=  0. 

12.  Find  the  locus  of  that  point  which  is  required  to  be 
equally  distant  from  the  points  (a,  b,  c)  and  (a',  h',  c'), 
Ans.  2(a'-  a)x  +  2(6'-  h)y  +  2(c'-  c)z 

=  a"  -  a'  +  b"  -b'  -\-  c"  -  c\ 


260  GEOMETRY  OF  THESE  DIMENSIONS. 

13.  If,  in  the  preceding  problem,  the  point  {a',  V ,  c')  is 
on  the  straight  line  from  the  origin  to  {a,  b,  c),  and  m  times 
as  far  from  the  origin  as  {a,  h,  c),  show  that  the  perpendicu- 


m  +  1 


lar  from  the  origin  upon  the  plane  is  — - —  Va"  -\- b"^  -\-  &. 

14.  The  plane  x  -\-  y  -\-  z  —  d  =  0  h  required  to  bisect 
the  line  from  the  origin  to  the  point  {a,  l,  c).  Find  the 
yalue  of  d.  Ans.     d  =  ^a  +  Z*  +  c). 

15.  Find  the  equation  of  the  plane  passing  through  the 
origin  and  through  the  line  of  intersection  of  the  planes 

2x  +  dy  +  4:Z  4'   2^  =  0; 
^  +    y  +    z  —  22J  =  0. 

Ans.     6x  -\-  7y  -\-  9z  =  0. 

16.  Find  the  equation  of  the  plane  which  shall  pass  through 
the  point  (2,  3,  5)  and  through  the  line  of  intersection  of  the 
two  planes 

X  -\-  y  -\-    2;  —  5  =  0; 
X  —  y-\-2z-{-l  =  0. 

A71S.     X  -{-  dy  —  11  =  0. 
Calling  the  two  expressions  P  and  P',  the  equation,  of  any  plane  pass- 
ing through  the  intersection  of  P  and  P'  may  be  written  in  the  form 
XP-\-P'  =0.     We  determine      by  the  condition  that  this  equation 
shall  be  satisfied  when  we  have  x  =  2,  y  =  S  and  2  =  5. 

17.  Write  the  equation  of  the  plane  passing  through  the 
origin  and  perpendicular  to  the  two  planes 

x-i-y-    z  =  0; 
X  —  y  —  2z  =  0. 

Ans.     Zx  —  y  -\-2z-=  0. 

18.  The  three  planes 

X—    2y  —    3z  =  0, 
2x-{-      y  —  nz  =  0, 
Vx  +  m'y  4-  7i'z  =  0, 
are  each  to  be  perpendicular  to  the  other  two.      Find  the 
least  integral  values  of  V,  m',  n'  and  n  which  satisfy  this  con- 
dition, and  thus  show  that  the  equations  of  the  second  and 
third  planes  are 

2a;  +  2/  =  0; 
Zx  —  ^  -\-hz=.  0. 


CHAPTER  III. 
THE  STRAIGHT  LINE  IN  SPACE. 


243.  Theokem  I.  The  position  of  a  line  is  completely 
determined  ly  its  projections  tipon  any  two  non-parallel 
planes. 

Proof.  Through  the  projection  on  one  of  the  planes  pass 
a  plane  ^  J_  to  that  of  projection.  The  line  projected  then 
must  lie  entirely  in  the  plane  R. 

In  the  same  way,  the  line  must  lie  entirely  in  the  plane 
S  Lio  the  other  plane  of  projection  and  containing  the  other 
projection.  Hence  the  line  is  the  intellection  of  the  planes 
R  and  S. 

There  can  be  only  one  plane  R  and  one  plane  S,  because 
along  a  given  line  in  a  plane  only  one  ±  plane  can  be  passed. 
Hence  there  is  but  one  line  in  which  these  planes  can  inter- 
sect, and  this  is  the  line  whose  projections  are  given.     Q.  E.  D. 

244.  Equations  of  a  Straight  Line.  Since  any  one 
equation  between  the  co-ordinates  of  a  point  represents  a  sur- 
face, at  least  two  equations  are  necessary  to  represent  a  line 
in  space.  These  equations,  considered  separately,  represent 
two  surfaces.  Considered  simultaneously,  that  is,  requiring 
the  co-ordinates  to  satisfy  them  both,  they  represent  the  line 
in  which  the  surfaces  intersect. 

The  most  simple  form  of  the  equations  of  a  straight  line 
are  given  by  the  equations  of  the  planes  in  which  it  is  pro- 
jected upon  any  two  of  the  co-ordinate  planes,  XZ  and  YZ 
for  example.     The  equation 

X  =^hz-\-  a 

(y  being  left  indeterminate)  represents  a  certain  plane  paral- 
lel to  the  axis  of  Y{%  231);  that  is,  the  co-ordinates  of  all  the 


262  GEOMETRY  OF  THREE  DIMENSIONS. 

points  in  this  plane  satisfy  the  equation,  and  vice  versa.  In 
the  same  way,  every  point  whose  co-ordinates  satisfy  the  equa- 
tion 

y  z=  Jcz  -\-  h 

lies  in  a  certain  plane  parallel  to  the  axis  of  X.  Hence 
every  point  whose  co-ordinates  satisfy  both  equations  must 
lie  in  both  planes,  that  is,  in  the  line  of  intersection  of  the 
planes.  The  two  equations  taken  simultaneously  therefore 
represent  a  straight  line. 

Eemark.  Any  two  consistent  and  independent  simulta- 
neous equations  between  the  co-ordinates,  for  instance, 

ax  -^-hy  -i-cz  -\- d  =  0,  )  ,^. 

a'x  +  I'y  -f  c'^  +  ^'  =  0,  f  ^  ^ 

equally  represent  a  straight  line,  namely,  the  line  in  which 
the  planes  intersect.     But  the  forms 

.  x  =  liz-\-aA  /«v 

y  =  hz^l,\  ('^ 

are  preferred  because  they  are  more  simple. 

We  also  remark  that  the  form  (1)  can  always  be  reduced 
to  the  form  (2)  by  first  eliminating  y  and  then  x  from  the  two 
equations. 

EXERCISES. 

1.  Express  the  equations  of  the  line  of  intersection  of  the 
planes 

3a;  -  2?/  +    z^  hd  =  0, 

-x-{-    y  -^  2z  -  4:d  =  0, 

in  the  form  (2). 

2.  Express  in  the  form  (2)  the  equations  of  the  three  lines 
of  intersection  of  the  planes 

X  —    y  —  z  =  a; 
X  -{-    y  —  z  =  b; 


,x  =  -  bz-Jr-  3d; 
Ans. 


THE  STRAIGHT  LINE  IN  SPACE.  263 

3.  Explain  how  it  is  that  the  equation  of  a  line  in  one  of 
the  co-ordinate  phines  (the  other  co-ordinate  being  supposed 
zero)  is  the  same  as  tlie  equation  of  the  plane  passing  through 
that  line  and  parallel  to  the  third  co-ordinate. 

4.  Prove  that  if  we  represent  the  equations  of  a  straight 
line  [(1)  or  (2),  for  example]  in  the  form 

P  =  0,         Q  =  0, 
then  the  equations 

mP  -\-  nQ  =  0,        mP  -  nQ  =  0, 
771  and  n  being  constants,  will  represent  the  same  line. 

245.  Symmetrical  Equations  of  a  Straight  Line.  The 
equations  of  a  straight  line  may  be  represented,  not  only  by 
two  equations  between  the  three  co-ordinates,  but  by  express- 
ing each  of  the  three  co-ordinates  as  a  function  of  a  fourth 
variable.     To  do  this,  let  us  put 

x^y  ^o>  ^o>  ^^^^  co-ordinates  of  any  fixed  point  of  the  line; 

X,  y,  z,  the  co-ordinates  of  any  other  point  of  the  line; 

p,  the  length  between  the  points  {x^,  y„,  z^)  and  {x,  y,  z). 

Then,  a,  §  and  y  being  the  angles  which  the  line  makes 
with  the  CO- ordinate  axes,  we  have,  by  §  215, 

a;  —  a;^  =  p  cos  a\  \ 

2/  -  ^0  =  P  cos  /?;  V  (3) 

2   —  ^^o   =  P  cos  /.  ) 

Here  x^,  y^,  z^,  a,  ^  and  y  are  supposed  to  be  constants 
which  determine  the  position  of  the  line  in  space,  while 
X,  y,  z  and  p  are  variables.  Assigning  any  value  we  please  to 
p,  we  shall  have  corresponding  values  of  x,  y  and  z,  which 
will  be  the  co-ordinates  of  that  point  P  on  the  line  which  is 
at  the  distance  pfrom  the  point  {x^,  y^,  z^).  Since  for  every 
point  on  the  line  there  will  be  one  and  only  one  value  of  p, 
and  for  this  value  of  p  one  value  and  no  more  of  each  co- 
ordinate, and  vice  versa,  the  equations  (3)  will  represent  all 
points  of  the  line,  and  no  others.  They  are  therefore  the 
equations  of  a  straight  line. 

These  equations  (3)  are  readily  reduced  to  the  form  (2)  by 


264  OEOMETBY  OF  THREE  DIMENSIONS. 

eliminating  p,  first  between  the  first  and  third,  and  then  be- 
tween the  second  and  third.     We  thus  find 

cos  a     .    x^  cos  y  —  z.  cos  a 

X  = z  +  — ; 

cos  y  cos  y 

cos  /3     ,    y.  cos  V  —  z.  cos  /3 

y  = -z  +  — -. 

^       cosy  cos  y 

The  equations  may  also  be  reduced  to  the  symmetric  form 


^-^0  _  y-Vo  _ 


cos  a        cos  /3        cos  y 

246.  Introduction  of  Direction- Vectors.  In  the  equa- 
tions (3)  we  may  introduce,  instead  of  the  direction-cosines, 
any  three  quantities  proportional  to  them,  without  changing 
the  line  represented  by  the  equation.  Let  these  quantities  be 
I,  m  and  n,  so  that  the  equations  become 

x  =  x^-\-lp;    J 

To  show  that  the  line  is  unchanged,  we  proceed  as  in  §  217, 
where  we  have  shown  that  the  proportionality  of  I,  m  and  n 
to  the  direction-cosines  may  be  expressed  by  the  equations 

=  (T  cos  a;        7n  =  ff  cos  /?;         n  =  G  cos  y. 
By  substitution  the  equations  (4)  become 
x  =  x^-\-  p6  COS  a\ 
y^y.^-P^  cos  /?; 
z  —  z^-\-  p<5  cos  y. 

These  equations  are  the  same  as  (3),  except  that  pa  takes 
the  place  of  p;  that  is,  the  distance  between  the  points 
(^o>  ^0?  ^o)  ^^^  (^>  y^  ^)  is  P^  instead  of  p.  Hence  the  systems 
(3)  and  (4)  represent  the  same  line,  except  that  in  (4)  p  repre- 
sents length  -^  a,  instead  of  length  simply. 

Cor.  We  may  multiply  the  three  direction-vectors  in  the 
symmetrical  equations  of  a  line  hyany  common  factor  without 
changing  the  line  represented. 


^^-  ^<^^^  ^-^/--^-^^ 


Jfu-r^^      ^IlZJI'      y.-'/^      -z  _2  __j? 


THE  STRAIGHT  LINE  IN  SPACE.  265 

Remark.  The  forms  (3)  and  (4)  have  a  great  advantage 
in  nearly  all  the  investigations  of  Analytic  Geometry,  and 
will  therefore  be  exclusively  employed.  The  advantage 
arises  from  the  fact  that  the  three  co-ordinates  which  fix  the 
position  of  some  one  point  of  the  line  are  completely  distinct 
from  the  quantities  I,  m  and  n  which  express  its  direction. 

EXERCISES. 

1.  Express  the  co-ordinates  of  the  three  points  in  which 
the  line  given  by  the  equations  (3)  intersects  the  three  co- 
ordinate planes  respectively.  Express  also  the  corresponding 
values  of  p. 

2.  Write,  in  the  form  (4),  the  equations  of  a  line  passing 
through  the  point  (a,  h,  c)  and  parallel  to  the  axis  of  Z. 

3.  Write,  in  the  same  form,  the  equations  of  a  line  passing 
through  the  point  {x^,  y^,  z^  parallel  to  the  plane  of  XJTand 
making  equal  angles  with  the  axes  of  Xand  Y. 

4.  Write  the  equations  of  a  line  passing  through  the  point 
(x^,  y^,  z^  and  making  equal  angles  with  the  co-ordinate 
planes.  Express  also  tlie  co-ordinates  of  the  three  points  in 
which  it  intersects  the  co-ordinate  planes. 

Ans.,  in  part.  It  intersects  the  plane  of  YZ  in  the  points 

y  =  yo  -  ^o'^      ^  =  ^0  -  ^0- 

5.  Show  that  the  equations  of  the  line  passing  through  the 
points  {x^y  y^f  z^)  and  (x^,  y^,  z^)  may  be  written  in  the  form 

X  =  x^  -\-  (x^  —  xjp;    f>^^-<  <^y<'^  ts  /^^  ,^^ 

y^y.^{y.-  y.)P',  ^r'^'T-  T'^ti^ 

state  to  what  distance  on  the  line  corresponds  the  unit  of  p 
in  these  equations,  and  find  the  co-ordinates  of  the  points  in 
which  the  line  intersects  the  co-ordinate  planes. 

6.  Write  the  three  symmetrical  equations  of  the  straight 
line  joining  the  points  (1,  1,  2)  and  (2,  3,  5).  Find  the 
angles  which  it  makes  with  the  co-ordinate  axes,  tlie  points 
in  which  it  intersects  the  co-ordinate  planes,  and  the  distances 
between  these  points. 


266  GEOMETRY  OF  THREE  DIMENSIONS. 

347.  Condition  that  a  Line  shall  he  ])arallcl  to  a  Plane. 
So  long  as  the  coefficients  I,  m  and  n  in  the  equations  (4)  of 
a  straight  line  are  entirely  unrestricted,  these  equations  may, 
by  giving  suitable  values  to  I,  m  and  n,  be  made  to  represent 
any  line  whatever  passing  through  the  point  (x^,  y^y  2; J.  If, 
however,  they  be  subjected  to  a  homogeneous  equation  of  con- 
dition, the  lines  will  be  restricted,  as  we  shall  now  show. 

Theorem  II.  If,  in  the  symmetrical  equations  of  a 
straight  line,  the  direction-vectors  m,  n  and  fp  are  required 
to  satisfy  a  linear  equation,  the  line  will  lie  in  or  he  j^nrallel 
to  a  certain  inlane. 

Conversely,  the  requirement  that  the  line  shall  lie  in  or  he 
parallel  to  a  certain  pla7ie  is  indicated  hy  a  linear  equation 
hetioeen  the  direction-vectors. 

Proof.  Let  the  linear  equation  which  w,  n  and  p  are  re- 
quired to  satisfy  be 

Al  +  Bm  -f  Cn  =  0.  {a) 

I  say  that  every  point  of  every  possible  line  represented  by  the 
equations  (4)  will  then  lie  in  the  ^olane  whose  equation  is 

Aix  -  X,)  +  B(y  -  y,)  +  C(z  -  z,)  =  0,  (h) 

and  will  therefore  be  parallel  to  every  plane  whose  direction- 
vectors  are  A,  B  and  C.  For,  by  multiplying  the  equations 
(4)  respectively  by  ^,  -S  and  G,  transposing,  and  adding  the 
products,  we  find 

A{x  -  X,)  +  B{y  -  y,)  +  C(z  -  z,)  =  (Al  +  Bm  +  Cn)p. 

I^ow,  by  hypothesis  (a),  the  second  member  of  this  equation 
vanishes.  Hence  all  values  of  the  co-ordinates  x,  y  and  z 
which  satisfy  (4)  also  satisfy  [h).  Hence  every  point  of  the 
line  lies  in  the  surface  whose  equation  is  (h),  and  this  surface 
is  a  plane,  by  §  229. 

Every  plane  whose  direction-vectors  are  ^,  B  and  G  is 
parallel  to  (h),  because  perpendicular  to  the  same  line.  Hence 
(a)  is  the  condition  that  the  line  (4)  is  parallel  to  every  such 
plane. 


THE  STRAIGHT  LINE  IN  SPACE.  267 

Next,  let  it  be  required  tluit  the  line  (5)  shall  lie  in  the 
plane  whose  equation  is 

Ax  +  Bi/  -^  Cz  -{-D=  0.  (c) 

I  say  that  the  coeflBcients  m,  n  and  f  must  satisfy  the  linear    X 
equation  ^ 

Al  +  Bm  +  Cn  =  0. 

For,  by  substituting  in  (c)  the  values  of  x,  y  and  z  from  (4), 
we  have 

Ax,  +  By,  +  Cz,  +  D  +  (M  +  Bm  +  Ch)p  =  0,     {d) 

which  equation  must  be  satisfied  for  all  values  of  p.  Now,  by 
hypothesis,  the  point  {x^,  y^,  z^  lies  on  the  line,  and  therefore 
lies  in  the  plane  (c)  which  requires  it  to  satisfy  the  equation 

-4^-0  +  %o  -^Cz,-\-D  =  Q. 

Hence,  in  order  that  the  equation  {d)  may  be  satisfied,  we 
must  have 

Al  +  Bm  +  Cn  =  0.  (5) 

248.  Common  Perpendicular  to  Two  Lines.  It  is  shown 
in  Geometry  that  two  non-parallel  lines  have  one  and  only  one 
common  perpendicular,  and  that  this  perpendicular  is  the 
shortest  distance  between  the  lines.  Let  us  now  solve  the 
problem, 

To  find  the  equation  of  the  common  perpendicular  to  two 
given  lines. 

We  shall  express  the  equations  of  the  given  lines  in  the 
form  (3),  putting,  for  brevity, 

^u  Pv  Yv  I  the  direction-cosines  of  the  ffiven  lines, 
^.>  A.  r.. ' 
and 

a,   /?,    ;/,  those  of  the  required  perpendicular. 
Thus  the  symmetrical  equations  of  the  given  lines  will  be 
x  =  x^-\-  a^p',  \  rx  =  x^-\-  a^p\ 

y  =  yr-\-  P^9\  \  and  j  7/  =  y^  -I-  ^^p; 

^  =  ^»  +  r,P;  '  Kz  ^  z,A,-  y^p. 


268 


OEOMETRT  OF  THREE  DIMENSIONS. 


Let  as  first  find  the  direction-cosines  a,  p,  y.     By  §§  216, 
217,  we  have  the  equations 


a^a  -f  fi^p  +  y^y  =  0; 


(6) 


Eliminating  first  yS  and  then  /from  these  equations,  we  have 

(a^p,  -  a^P^)a  +  (P^y^  -  p^y^)y  =  0; 
{y,a,  -  y^a^)a  +  (/3,y^  -  /?^;/J/?  =  0. 

Dividing  these  equations  by  ay  and  a/3,  respectively,  gives 

or  /?        ~        r       ~ 

/^^  =  Ar. -AKi; 

Mr  =  «^iA  -  «^2;^i- 

Taking  the  sum  of  the  squares  of  these  equations, 

M'  =  (Ar,  -  ArJ'  +  (r.«.  -  n^^Y  +  («.A  -  «,A)^  (7) 

which  is  the  square  of  the  sine  of  the  angle  between  the  given 
lines  (§  218). 

The  direction-cosines  a,  /?  and  y  are  therefore 


a  =  —^-^ 


Ar,. 


sin  V 


/3=y^ 


n^^. 


sm  V 


sin  ^' 


(8) 


t^  being  the  angle  between  the  given  lines.  Thus  the  direc- 
tion of  the  required  line  is  completely  determined. 

To  complete  the  solution,  we  must  find  the  co-ordinates 
of  some  point  of  the  line.     Let  us  then  pat 

(a,  b,  c)  the  point  in  which  the  required  line  intersects  the 
first  of  the  given  lines.  The  equations  of  the  required  line 
may  then  be  written 


THE  STRAIGHT  LINE  IN  SPACE.  269 

x=  a  -{-  ap;  \ 

z  =c-\-yp.) 

Let  us  also  put 

p„  the  distance  of  the  point  (a,  by  c)  from  (x^,  ?/,,  z^)  on 
the  first  given  line  ; 

p^,  the  distance  from  {x^,  y^,  z^)  on  the  second  given  line 
to  the  point  in  which  the  required  line  intersects  it; 

Po,  the  distance  of  the  points  of  intersection,  that  is,  the 
length  of  the  shortest  line  between  the  given  lines. 

Then,  equating  the  expressions  for  the  co-ordinates  of  the 
points  of  intersection  on  the  two  lines,  we  have  the  six  equa- 
tions 

a  =  x^-\-  «iPi;  )  Intersection  of  required 
b  =2/1+ APiJ  (  ^ii^6  with  first  given 
c  =  2;,  -f  y^p^.  )      line. 


a  -f-  ^Po  =  ^2  +  ^iPil  )  Intersection  of  required 
b  -\-  /?Po  =  2/2  +  A/^aJ  f  ^^^^  wi^^  second  given 
c -\- ypo  =  ^, -\-  r,P,'  ^      line. 


(c) 


These  six  equations  suffice  to  completely  determine  the  six 
unknown  quantities,  a,  b,  c,  p^,  p,,  p^.  First  subtracting 
corresponding  equations  in  the  two  sets,  we  eliminate  a,  b  and 
c,  and  have  three  equations  which  we  may  write  in  the  form 

^Po    +     ^iPl     -     ^2P.     =     ^1     -     ^,»    ) 

M  +  ^^p^  -  Ap.  =  y.-  y^^  \  W 

yPo  +  r^p^  -  r,p,  =  ^,  -  ^^, ' 

and  which  contain  only  the  three  unknown  quantities  p^,  p^ 
and  Pj.  Multiplying  the  equations  in  order  by  a,  /?  and  y, 
taking  their  sum  and  referring  to  the  relations  (6),  we  have 

po  =  ^K  -  ^i)  +  ^(y.  -  y,)  +  rfe  -  ^J-         (9) 

From  the  manner  in  which  p^  has  been  defined,  it  is  equal 
to  the  shortest  distance  between  the  two  lines  (1)  and  (2),  be- 
cause it  is  the  distance  from  the  point  [a,  b,  c)  to  the  point 
in  which  the  shortest  line  intersects  the  second  line. 


270  GEOMETRY  OF  THREE  DIMENSIONS. 

If  we  substitute  for  or,  /?  and  y  their  values  (8),  we  have 


(10) 


Again,  multiplying  the  equations  (d)  by  a„  y^i  and  ;^i,  and 
adding,  and  then  by  a^,  /?,  and  ;/„  and  adding,  we  find 

Pr  -  P.  cos  V  =  a^x^  -  x^  4-  /?,(?/,  -  ?/J  4-  ;/X2^2  -  ^i)  =  ^; 
/9,  cos  V  -  p,  =  a,(a:,  -  a:,)  +  /?,(y,  -  ?/J  +  ;/,(2;,  -  z^  =  r,. 

Hence 

_  ^\  ~  '^\  cos  V 
^'  ~        sm^       ' 

r,  cos  V  —  r. 

P^  =  - — —2 ^• 

^  sin  V 

To  find  a,  b  and  c,  we  have  only  to  substitute  the  value  of 
p,  in  (c),  which  gives 


,    a,r,  —  ar^  cos  v 

a  =  .T,  +  --5-5^ — H-^ ; 

'  sin  V 

^  ~  ^^  "^  sin'  V 

.   K,^',  —  Ki^o  cos  V 

C  z=z  Z.   -\-  ^-^ /-f-^ . 

'  sin^  V 


(11) 


The  values  of  a,  Z>,  c,  a,  y^and  y  in  (11)  and  (8)  being 
substituted  in  (Z>),  the  equations  of  the  shortest  line  are  com- 
plete. 

249.  Condition  of  Intersection.  Since  p^  in  (9)  expresses 
the  shortest  distance  between  the  two  lines,  the  condition 
that  the  lines  shall  intersect  is  found  by  putting  p  —  0. 
Substituting  for  a,  ft  and  y  their  values  (8),  this  condition 
gives 

If,  instead  of  or,  ft  and  y,  we  use  the  quantities  Z,  m  and  n, 
we  must,  from  the  proportionality  of  these  factors  to  a,  ft  and 
)/,  have  a,  ft  and  )/  equal  respectively  to  /,  m  and  w,  each 
multiplied  by  the  same  factor. 


THE  STRAIGHT  LINE  IN  SPACE.  271 

If  we  call  p  and  q  these  factors  in  the  cases  of  or,,  /?„  y^ 
and  «,,  /?„  ;^,  respectively,  we  have 

and  the  condition  of  intersection  becomes 

(w,7la—Wani)(ira-a'i)+(;iiZ2  — 712^1)  (ya-2^i)-f(ZiWa-^2W,)  (2a— Si)=0.     (12) 

250.  Problem.  To  find  the  point  in  which  a  lirie  m- 
tersects  a  surface. 

Since  the  point  of  intersection  lies  on  the  line,  there  will 
be  a  definite  value  of  p  corresponding  to  it.  This  value  of  p, 
being  substituted  in  the  equation  of  the  line,  will  give  values 
of  the  co-ordinates  x,  y  and  z  which,  if  p  is  properly  taken, 
will  satisfy  the  equation  of  the  surface.  We  therefore  proceed 
as  follows: 

Calling,  for  the  moment,  {a,  b,  c)  any  one  point  of  the 
given  line,  we  substitute  in  the  equation  of  the  surface,  for 
X,  y  and  z,  the  expressions 

X  =  a  -\-  Ip',        y  =zh  -\-  mp;        z  =  c  -{-  np.  {a) 

The  equation  of  the  surface  will  then  contain  no  unknown 
quantity  except  p,  and  is  to  be  solved  so  as  to  get  an  expres- 
sion for  p  which  shall  satisfy  it. 

This  expression  being  substituted  in  the  equations  {a) 
will  give  the  required  values  of  the  co-ordinates  of  the  point 
of  intersection. 

If  the  equation  in  p  is  of  a  higher  degree  than  the  first, 
there  will  be  several  values  of  p,  and  therefore  several  points 
of  intersection. 

Example.     Find  the  point  in  which  the  line 

a;  =  2  +  2p, 
3/  =  3  -  2p, 
2;  =  5  —  p, 
intersects  the  plane 

2a;  -  3y  -  2J  -f  8  =  0. 


272  OEOMETRT  OF  THREE  DIMENSIONS. 

Substituting  the  values  of  the  co-ordinates  in  the  equa- 
tion of  the  plane,  we  find 


which  gives  P  —  T\ 


10  +  lip  +  8  =  0, 
2 
11' 


whence 

^  =  2A>      y  =  2t\,      z  =  4^, 

are  the  co  ordinates  of  the  point  of  intersection. 

The  same  general  method  applies  whenever  points  fulfil- 
ling any  condition  whatever  are  to  be  found  on  one  or  more 
lines.  Each  line  must  have  its  own  value  of  p,  which  we 
may  distinguish  from  the  values  for  other  lines  by  accents  or 
subscript  numbers.  The  values  of  the  co-ordinates,  expressed 
in  terms  of  p,  are  to  be  substituted  in  each  condition,  and 
equations  with  the  p's  as  the  only  unknown  quantities  will 
thus  be  formed. 

EXERCISES. 

1.  Write  the  equations  of  the  sides  of  the  triangle  whose 
vertices  are  at  the  points  (1,  2,  3),  (3,  2,  1)  and  (2,  3,  1),  and 
find  the  angles  of  the  triangle. 

Ans.,  171  part.  30°,  60°,  90°. 

2.  Find  the  points  in  which  the  line  joining  the  points 
(1,  2,  3)  and  (2,  3,  4)  intersects  the  co-ordinate  23lanes. 

Ans.  (0,  1,  2);     (-  1,  0,  1);     (-  2,  -  1,  0). 

3.  Write  the  symmetrical  equations  of  the  line  passing 
through  the  point  (a,  b,  c)  and  perpendicular  to  the  plane 

px  -{-  qy  -\-  rz  =  0. 

4.  An  equilateral  triangle  has  one  vertex  in  each  co-ordi- 
nate plane,  at  the  distance  h  from  each  of  the  axes  lying  in 
that  plane.  Write  the  equations  of  each  of  its  sides,  taking 
the  middle  point  of  each  side  as  the  point  from  which  p  is 
measured. 

Ans.,  in  part,  x  =  h;  )  Equations 

y  =:^h  -\-  p;y    of  one  of 
z  =ih  —  p.)    the  sides. 


THE  STRAIGHT  LINE  IN  SPACE.  273 

5.  Ill  what  points  does  the  line  of  intersection  of  the  two 
planes 

x-{-ij  -z    ^7, 
x-y-\-2z  =  l, 

intersect  the  co-ordinate  planes? 

Ans.   (4,3,0);     (0,15,8);     (5,0,-2). 

6.  Express  the  point  in  which  the  line  (4),  §  246,  inter- 
sects the  plane  Lx  -f-  My  -{-  Nz  =  0. 

A         '  i  M{inx^  —  ly^)  +  N(nx^  —  Iz,) 

Ans.,  m  part,  x  =  — ^^ — Vv    .    \r — ,    ,x -• 

^  lL-\-  Mm  +  Nn 

7.  Write  the  symmetrical  equations  of  the  line  of  inter- 
section of  the  two  planes 

a;  +  2?/  —  30  —  5  =  0, 
2a;-?/    +22^  +  7  =  0, 

taking  as  the  zero  point  of  the  line  that  in  which  it  intersects 
the  plane  XY. 

9  17 

Ans.  X  =  —  ——  p',      y  z=—-  -\-  8p;     z  =  0  -{-  6p. 

O  0 

In  cases  where  direction-vectors  appear  as  unknown  quantities  in 
equations,  there  will  be  but  two  equations  for  the  three  vectors.  In  this 
case  we  determine  any  two  in  terms  of  the  third,  and  assign  to  the  lat- 
ter such  value  as  will  give  the  simplest  form  to  the  results. 

8.  Write  the  equations  of  the  line  passing  through  the 
point  (3,  1,  5)  and  intersecting  the  axis  of  X  perpendicularly. 

9.  Write  the  equations  of  the  line  passing  through  the 
point  {a,  l,  c)  and  parallel  to  each  of  the  planes 

x-\-y-^z  =  0', 
X  —  y  -\-  z    =0. 

Ans.  X  =  a  -{-  p; 
y  =  h-{-Sp; 
z  =  c  -{-  2p. 

10.  Find  the  condition  that  the  line  (4),  §  246,  shall  inter- 
sect the  axis  of  Z.  Ans.  mx^  =  ly^. 

The  condition  requires  that  the  points  in  which  the  line  intersects  the 
planes  of  XZ  and  YZ  respectively  shall  be  the  same,  that  is,  correspond 
to  the  same  value  of  p. 


274  GEOMETRY  OF  TEHEE  BIMEJS8I0KS. 

11.  Find  the  equation  of  the  line  passing  through  the 
point  (5,  2,  4)  and  intersecting  perpendicularly  that  line 
through  the  origin  whose  direction-vectors  are  I  =  1,  m  =  2, 
n  =  2.  Ans.  x  =  5  —  14p;     y  =  2  -\-  8/3;     z  =  4:  —  p. 

Write  the  symmetrical  equations  of  the  given  and  the  required  line, 
calling  p'  the  variable  for  the  one  line,  and  p  for  the  other.  The  con- 
dition that  some  one  point  {x,  y,  z)  shall  satisfy  the  equations  of  both 
lines  then  gives  three  equations  of  condition  between  I,  m,  n,  p'  and  p, 
and  the  condition  of  perpendicularity  gives  a  fourth, 

12.  Deduce  the  condition  that  two  lines  shall  intersect  by 
the  principle  that  there  must  then  be  07ie  set  of  values  of  x,  y 
and  z  which  shall  satisfy  the  equations  of  both  lines,  these 
values  of  the  co-ordinates  being  given  in  terms  of  one  value 
of  p  for  the  one  line,  and  another  value  for  the  other  line. 

The  condition  gives  three  equations  between  the  two  quantities  p 
(on  one  line)  and  p'  (on  the  other),  and  the  values  of  p  and  p'  must  be 
the  same,  whether  we  derive  them  from  one  or  another  pair  of  the  equa- 
tions. 

13.  Write  the  equation  of  the  plane  which  contains  the 
two  intersecting  lines 


x  =  a-]-  p\ 

x  =  a-  p; 

y=zh-p; 

y  =  d-2p; 

z  =c-2p; 

z  =c  -3p. 

A71S.  X  —  6y  -\-  Sz  —  a  -\-  5b  —  3c  =  0, 

Note  that  the  condition  that  a  plane  shall  contain  or  be  parallel  to  a 
line  is  the  same  as  that  a  line  shall  be  parallel  to  or  lie  in  a  plane. 

14.  Find  that  plane  which  is  parallel  to  each  of  the  lines 

X  =  a  —  2p;  X  =  a^  -\-  p; 

y  =  b-p\  y=zb'^2p', 

z  =  c  -\-  p;  z  =  c'  —  p; 

and  equidistant  from  them.     Also  find  the  common  distance. 
A?is.   2x  -\-  2y  -\-  6z  —  a  -  a'  —  b  —  y  —  Sc  —  3c'  =  0. 

Common  dist.   =  ^ -«' +  ^  -  ^1+ 3(c  -  O 

2i/ll 


CHAPTER  IV. 

QUADRIC   SURFACES. 


General  Properties  of  Quadrics. 

251.  Def.  A  quadric  surface  is  the  locus  of  a  point 
in  space  whose  co-ordinates  are  required  to  satisfy  an  equation 
of  the  second  degree. 

Remark.     A  quadric  surface  is  called  a  quadric  simply. 
The  most  general  form  of  an  equation  of  the  second  degree 
between  the  co-ordinates  is 

gx^  +  hy^  +  hz"  +  2g'yz  +  Wzx  +  Wxy 

+  2g"x  +  W'y  +  Wz  -\-d  =  Q,     (1) 

because  the  terms  in  this  equation  include  all  powers  and 
products  of  the  co-ordinates  x,  y  and  z  up  to  those  of  the 
second  degree. 

The  number  of  coefficients,  as  written,  is  ten.  But  since, 
by  division,  we  may  reduce  any  coefficient  to  unity,  their 
number  is,  in  effect,  equivalent  to  nine.     Hence: 

Theorem  I.  Nine  conditions  are  necessary  to  determine 
a  quadric  in  space. 

Eemark.  In  discussing  the  equation  (1),  we  regard  the 
coefficients^,  h,  h,  g' ,  etc.,  as  given  constants,  unless  other- 
wise expressed. 

We  may  trace  out  certain  analogies  between  the  quadric 
and  conic,  by  treating  tlie  former  in  the  same  general  manner 
in  which  we  treated  the  latter  in  Part  I. 


276  OEOMETRT  OF  THREE  DIMENSIONS. 

252.  hiter sections  of  a  Q^iadric  with  a  Straight  Line, 
Let  the  equations  of  the  line  be 

^  =  ^0  +  ^pn 

y  =  yo  +  ^^ip;  r  («) 

Z  =  Z^-i-    7ip.  ) 

The  problem  now  is,  What  values  of  x,  y  and  z  satisfy  both 
these  equations  and  (1)?  (Of.  §  250.)  If  we  substitute  these 
expressions  for  x,  y  and  z  in  (1),  thus: 

gx'^=g(x:^nx,p^rfy^), 
etc.  etc., 

we  shall  have  an  equation  in  which  all  the  quantities  except 
p  will  be  supposed  given.  Hence  p  can  be  determined  from 
the  equation.  When  this  is  done,  the  values  of  the  co-ordi- 
nates of  the  point  or  points  of  intersection  are  found  by  sub- 
stituting the  values  of  p  in  {a).  Now  the  equation  in  p  will 
be  of  the  second  degree,  and  will  therefore  have  two  roots, 
which  may  be  real,  imaginary  or  equal.     Hence: 

Theokem  II.  Every  straight  line  intersects  a  quadric  in 
two  real,  imaginary  or  coincident  poi^its. 

253.  Centre  of  Quadric.  Let  us  next  change  the  origin 
to  a  point  whose  co-ordinates  have  the  arbitrary  values  A,  B, 
G,  If  we  distinguish  the  co-ordinates  referred  to  the  new 
system  by  accents,  we  shall  have  (§219) 

X  ^  x'  -\-  A\ 
y=y'-\-B', 
z  =  z'  +  0, 

Substituting  these  values  in  the  general  equation  (1),  it  be- 
comes 

gx'""  +  7iv"  +  hz"  +  2g'y'z'  +  Wz'x'  +  Wx'y' 
+  Kg  A  H-  k'B  +  h'C  +  g")x' 
+  KhB  J^g'C^k'A-{-  h")y' 
-f-2(^•(7-f  A'^  -\-g'B  +  k")z' 
+  gA'  +  hB'  ^TcC  +  %g'BC^  WOA  +  WAB 

+  2^"^  +  WB  -f  U"  O-\-d  =  0.      (1') 


qUADRIC  SURFACES.  211 

Let  us  now  determine  the  co-ordinates  A,  B,  C  ot  the  new 
origin  by  the  condition  that  the  coefficients  of  x%  y'  and  z' 
shall  all  vanish.     To  do  this  we  have  to  solve  the  equations 

gA    J^lc'B^li'C=  -cj";\ 

h'A  -\-hB   -{-g'C  ^  -  //';  \  (2) 

¥A  +  g'B  +  y^C  =  _  Ic";  ) 

regarding  A,  B  and  G  as  the  unknown  quantities.  Since 
there  are  as  many  equations  as  unknown  quantities,  the  solu- 
tion will,  in  general,  be  possible. 

Let  us  now  suppose  the  equations  (2)  to  be  solved,  and  the 
resulting  values  of  A,  B  and  C  to  be  substituted  in  (1').  Let 
us  also  put 

d! ,  the  absolute  terms  in  (1'). 

Then,  omitting  accents,  the  equation  (1')  of  the  quadric  re- 
duces to 

gx^  4-  Tiy"  +  hz^  +  ^'yz  +  %li'zx  +  %¥xy  +  ^'  =  0.     (3) 

From  this  equation  we  may  deduce  a  second  fundamental 
property  of  the  quadric.  If  {x,  y,  z)  be  any  values  of  the  co- 
ordinates which  satisfy  (3),  it  is  evident  that  (—  x,  —  y,  —  z) 
will  also  satisfy  it.  Hence,  if  one  of  these  points  is  on  the 
quadric,  the  other  will  also  be  on  it.  But  the  line  joining 
these  points  passes  through  the  origin,  and  is  bisected  by  the 
origin,  that  is,  by  the  point  whose  co-ordinates,  referred  to  the 
original  system,  are  (A,  B,  C).  Since  {x,  y,  z)  may  be  any 
point  on  the  quadric,  we  conclude: 

Theorem  III.  For  every  quadric  there  is,  tn  general,  a 
point  loliicli  bisects  every  chord  passing  through  it. 

Def.  That  point  which  bisects  every  chord  passing 
through  it  is  called  the  centre  of  the  quadric. 

A  chord  through  the  centre  is  called  a  diameter  of  the 
quadric. 

354.  Section  of  a  Quadric  hy  a  Plane.  To  investigate 
the  equation  of  the  plane  curve  in  which  any  plane  intersects 
the  quadric,  we  may  take  a  pair  of  co-ordinate  axes  in  the 
cutting  plane.  This  we  do  by  simply  transforming  the  equa- 
tion  to  one  referred  to  new  axes;  and  we  may,  in  the  first 


278  OEOMETRT  OF  THREE  DIMENSIONS. 

place,  leave  the  origin  unchanged.     Accenting  the  new  co- 
ordinates, the  equations  of  transformation  will  be  (§220) 

x  =  ax'     -f  Py'     4-  yz'-, 
y  =  a'x'    +  ^'y'    +  y'z'-, 
%  =  a'V  +  y3"y'  +  ;/'V; 

^f  A  Vf  etc.,  being  the  direction-cosines  of  the  new  co-ordi- 
nate axes  relatively  to  the  old  ones. 

Now  when  we  substitute  these  expressions  in  the  general 
equation  (1)  and  arrange  the  terms  according  to  the  powers 
and  products  of  x',  y'  and  z',  we  shall  have  a  new  equation  of 
the  same  form  as  (1),  that  is,  one  containing  terms  in  a:", 
y'^y  /%  x'y',  etc. ;  the  only  change  being  that  the  coefiScients 
gy  h,  h,  g'y  etc.,  have  new  values.  We  may  therefore,  without 
any  loss  of  generality,  take  the  equation  (1)  as  representing 
the  transformed  equation,  and  consider  the  section  of  the  sur- 
face which  it  represents  by  a  plane  parallel  to  any  one  of  the 
co-ordinate  planes,  XY  for  example.  Let  us  then  suppose 
z=  c  m  (1).  The  equation  of  the  section  of  the  quadric  by 
the  plane  z  =  c  will  be,  omitting  the  accents, 

gx'  +  hf  +  Wxy  -f  2{h'c  +  g")x  +  ^g'c  +  h")y 

+  Tcc^  +  Wc  +  J  =  0. 

This  is  the  equation  of  a  conic  section.     Hence: 

Theorem  IV.  Every  plane  section  of  a  quadric  is  a  conic 
section. 

It  is  algt)  shown  in  §  198  that  all  conies  whose  equations 
have  the  same  coefficients  in  x^,  xy  and  y"^  are  similar  and 
similarly  placed.  Now,  in  the  above  equation,  the  coefficients 
g,  li  and  2^  remain  unaltered,  however  c  may  change;  that  is, 
however  we  may  change  the  position  of  the  cutting  plane,  so 
long  as  it  remains  parallel  to  the  plane  of  XY.     Hence: 

Theorem.  V.  All  sections  of  a  quadric  ly  parallel 
planes  are  similar  conies  and  have  their  principal  axes 
parallel. 

Cor.  If  any  plane  section  of  a  quadric  is  a  circle,  all 
sections  parallel  to  it  are  circles. 


QUADBIC  SURFACES.  279 

EXERCISES. 

1.  Find  the  centre  of  the  quadric 

x^  +  /-^^^  +  2;'  +  nyz  -\-  mx  =  0. 

2.  Write  the  equation  of  the  locus  of  the  point  required 
to  be  equally  distant  from  the  origin  and  from  the  plane 

ax-\:fiy^yz-p^O.       K  +  ^^  +  r^  =  l.)    §^~ 

3.  Write  the  equation  of  the  locus  of  the  point  equally 
distant  from  the  origin  and  from  tlie  plane 

CX-\-c'tJ  -p=  0,  (6'^  +  c''  =  1.) 

and  show  that  its  centre  is  at  infinity. 

255.  Conjugate  Axes  and  Planes.  Consider  this  prob- 
lem: 

To  find  the  locus  of  the  middle  points  of  all  chords  of  a, 
quadric  parallel  to  any  fixed  line,  and  therefore  to  each  other. 

Let  the  equation  of  any  one  of  the  chords  be 

X  =  x^  +  Ip', 

y  =  !/o  +  ^^p; 

z  =  z^  -^np. 

If  I,  m  and  n  remain  constant,  then,  by  asigning  all  values  to 
x^,  y^  andz^,  these  equations  may  represent  any  system  of  lines 
parallel  to  each  other.  Now,  we  find  the  two  points  in  which 
any  one  of  these  lines  intersects  the  surface  by  the  process  of 
§  250;  namely,  we  put  in  the  equation  (3)  of  the  surface 

x'^  =  x,'  +  2lx,p-^rp^;. 

y'  =  y'  +  2^^2/oP  +  '?^>'; 

z"  =zj'  -\-27iz^p  +  n'p'; 

y^  =  yo^o  +  (^^^0  +  '^^^o)P  +  tnup""', 
zx  =  z^x^  4-  {nx^  +  lz^)p  +  nlp^', 

^y  =  ^,y.  +  (^2/0  +  ^^o)p  +  i'nip\ 

For  brevity,  let  us  represent  the  result  of  substituting  these 
values  in  (3)  in  the  form 

A^Bp-{-  Cp'  =  0.  (a) 


280  GEOMETRY  OF  THREE  DIMENSIONS. 

Now,  we  may  choose  for  (a;„,  y^,  z^)  any  point  on  the  chord. 
Let  us  choose  the  middle  point.  This  point  will  be  deter- 
mined by  the  condition  that  the  two  values  of  p  from  the 
quadratic  equation  {a)  shall  be  equal,  with  opposite  algebraic 
signs.  The  condition  for  this  result  is  ^  =  0.  That  is,  writ- 
ing for  B  its  value,  the  condition  will  be 

glx,  +  hmy,  +  knz,  +  g^ny,  +  7nz,)  +  h'(7ix,  +  Iz,) 

+  k\ly,  +  mx,)  =  0. 

This,  then,  is  the  equation  which  the  middle  point  {x^,  y^,  z^) 
must  satisfy  as  x^,  y^  and  z^  vary.  Being  of  the  first  degree, 
it  is  the  equation  of  a  plane,  and,  having  no  absolute  term, 
the  plane  passes  through  the  origin,  that  is,  the  centre 
of  the  quadric.     Hence: 

Theorem.  VI.  The  locus  of  the  7nicldle  points  of  a  system 
of  parallel  chords  of  a  quadric  is  a  plane  through  the  centre. 

Def  A  plane  through  the,centre  of  a  quadric  is  called  a 
diametral  plane. 

That  diametral  plane  which  bisects  all  chords  parallel  to  a 
diameter  is  said  to  be  conjugate  to  such  diameter,  and  the 
diameter  is  conjugate  to  the  plane. 

That  diameter  whose  direction-vectors  are  I,  m,  n,  that  is, 
whose  equations  are 

X  =  Ip, 
y  —  mf>, 
z  =  np, 

will  be  called  the  diameter  {},  m,  n). 

Remark.  If  we  call  any  diameter  ^,  we  may  call  the 
conjugate  diametral  plane  A'. 

The  above  equation  of  the  diametral  plane  may  be  written 
out  thus,  the  subscript  zeros  being  omitted: 

{gl  -j-  ¥m  +  ¥n)x  +  {¥1  +  hm  -}-  g'7i)y 

+  {hn  4-  g'm  +  hn)z  =  0.       (4) 
That  is,  this  plan^  is  conjugate  to  the  diameter  {I,  m,  n). 

Theorem.  VI J.  If  a  diameter  B  lie  in  a  plane  A\  the 
conjugate  diametral  plane  B'  will  contain  the  diameter  A, 
conjugate  to  A'. 


QUADRIC  SURFACES.  281 

Proof.  Let  the  equation  (4)  represent  the  diametral 
plane  A',  and  let  the  diameter  B  be  (A,  //,  v).  By  §  24:7,  tlie 
condition  that  this  diameter  shall  lie  in  the  plane  (4)  is 

{gl  -f    h'm  -\-h'n)X  +  {k'l  -f  lim  +  g'n)}j. 

+  {h'l  +  g'm  +  /(;?0^  =  0, 
or,  rearranging  the  terms, 

{gX  4-  F//  +  ^V)?  4-  {k'X  +  7i//  +  ^V)77i 

+  {h'X  4-  ^'yw  +  ky)n  =  0. 

But  (§  247)  this  is  the  condition  that  the  diameter  (/,  m,  n), 
or  A,  shall  lie  in  the  plane 

(#  +  ^V  +  h'r)x  +  (FA  +  hpi  +  ^V)?/ 

which,  by  comparison  with  (4),  is  seen  to  represent  the  plane 
conjugate  to  the  diameter  (A,  yw,  r),  or  B;  that  is,  the  plane 
B\     Hence  this  plane  contains  the  diameter  A.     Q.  E.  D. 

Scholium.  Having  two  conjugate  diameters,  A  and  By 
with  their  diametral  planes.  A'  and  B',  arranged  as  in  this 
theorem,  the  intersection  of  the  planes  A'  and  B'  will  deter- 
mine a  third  diameter,  which  we  may  call  C.  Then,  because 
C  lies  in  both  the  planes  A'  and  B',  its  conjugate  plane  (7' 
will,  by  Theorem  VII.,  pass  through  both  A  and  B.  Thus 
we  shall  have  a  system  of  three  diametral  planes  whose  inter- 
sections will  form  three  diameters,  and  each  plane  will  bisect 
all  chords  parallel  to  its  conjugate  diameter. 

These  three  lines  and  planes  are  called  a  system  of  con- 
jugate axes  and  diametral  planes. 

2^Q,  Change  in  the  Direction  of  the  Axes.  To  simplify 
the  equation  (3)  still  further,  let  us  change  the  direction  of 
the  axes  of  co-ordinates,  leaving  the  origin  at  the  centre. 
This  we  do  by  the  substitution 

X  =  ax'  4-  fty'  4"  yz'', 
y  =  a'x'  4-  /3'y'  -\-  y'z*\ 
z  =  a'V  4-  /?"?/'  4-  y''z\ 

If  we  substitute  these  values  in  (3),  we  shall  have  an  equation 


282  GEOMETRY  OF  THREE  DIMENSIONS. 

the  terms  of  wliicli  we  can  arrange  according  to  the  powers 
and  products  of  x',  y'  and  2';  namely, 

^'\  y'\  z'\  y'z',  z'x\  x'y\ 
We  then  suppose  the  values  of  the  direction-cosines  a,  /?,  y^  a\ 
etc.,  to  be  so  determined  that  tlie  coefficients  of  y*7J ,  z*x'  and 
x'y'  shall  all  three  vanish.  This  will  require  three  equations 
of  condition  to  be  satisfied,  which,  with  the  six  relations  (14) 
of  §220,  will  completely  determine  the  nine  direction-cosines.* 
These  cosines  being  determined,  the  coefficients  of  a;"*,  y'"^ 
and  z'"^  will  all  become  known  quantities,  while  the  products 
y*z',  etc.,  will  disappear.  Thus,  omitting  once  more  the 
accents  from  the  co-ordinates,  the  equation  (3)  will  be  re- 
duced to  the  form 

Vx""  -\-m'y-  ■\-  n'z"  ^  cV  ^^, 

-y-ff-p  =  l,  (5) 

V,  w'  and  71'  being  known  quantities,  functions  of  the  origi- 
nal coefficients  in  (1).  It  will  be  seen  that  the  absolute  term 
d^  remains  unaltered  by  this  transformation. 

The  several  quantities  -7,,  -r,-,  etc.,  may  be  either  positive 

or  negative,  according  to  the  values  of  the  coefficients  wiiich 
enter  into  the  original  equation  (1). 

257.  Principal  Axes.     Let  us  put  A  for  the  value  of 

d'  .  ^' 

-,-  taken  positively;  the  first  term  of  (5)  will  be  ±  -:,  accord- 

ing  to  whether  it  is  positive  or  negative.     If  then  we  put 
a  =  VI=V±  J, 

the  term  will  become  ±  -..- 
a' 

*  The  equations  obtained  in  this  way  are  too  complex  for  con- 
venient management,  and  the  actual  values  of  the  direction-cosines 
must  be  found  by  the  differential  calculus,  or  by  an  application  of  the 
algebra  of  linear  substitutions.  "We  must  therefore,  at  present,  be  con- 
tented with  showing  the  possibility  of  the  solution,  which  is  all  that  is 
necessary  for  our  immediate  purpose. 


QUAD  RIG  SURFACES.  283 

In  the  same  way  the  other  terms  can  be  rcdnced  to  the 
form  ±  'jif  and  ±  -, .  Thus  the  general  equation  of  the  quad- 

0  0 

ric  can  finally  be  reduced  to  the  form 

±~±--fr±-,  =  l.  (6) 

a         b-        &  ^  ' 

Def.  The  quantities  a,  h  and  c  in  this  equation  are  called 
the  principal  axes   of  the  quadric. 

The  Three  Classes  of  Quaclrics. 

258.  There  are  now  four  possible  cases,  omitting  the 
exceptional  ones  in  which  a,  h,  or  c  is  zero  or  infinity. 

Case  I.  The  coefficients  of  the  first  member  of  (6)  all 
positive. 

Case  II.  Two  coefficients  positive  and  one  negative. 

Case  III.  One  coefficient  positive  and  two  negative. 

Case  IV.  All  the  coefficients  negative. 

In  the  last  case  no  real  values  of  the  co-ordinates  can  sat- 
isfy the  equation,  because  the  terms,  being  themselves  squares, 
are  essentially  positive,  and  therefore  with  the  minus  signs 
essentially  negative.  Hence  there  can  be  no  real  surface  to 
represent  the  equation.  But  in  the  other  three  cases  there 
will  be  real  loci.     Hence 

Tliere  are  three  rjeneral  classes  of  real  quadrics. 

259,  Class  I.  The  EUiiJsoid.  In  Case  I.  the  equation 
is 

--  4-  ^-  +  --  =  1  (7) 

Let  us  first  investigate  the  limiting  values  of  the  co-ordi- 
nates.   "Writing  the  equation  in  the  form 

^  I   l'  -  1  _  5 

we  see  that  when  —  c  >  2;  >  -\-^c,  the  co-ordinates  x  audi/  can- 
not both  be  real.  Hence  the  surface  is  wholly  included  between 
the  two  planes  ^  =  -f  ^  a^'^d  z  =.  —  c. 


284 


GEOMETRY  OF  THREE  DIMENSIONS. 


In  the  same  way  it  is  shown  that  the  surface  is  included 
between  tlie  planes  x  =^  -\-  a  and  x  —  —  a,  and  also  between 
the  planes  y  —  -\-h  and  y  =i  —h.  Hence  it  is  bounded  in 
every  direction. 

Because  its  sections  by  a  plane  are  of  the  second  order, 
and  limited  in  extent,  they  must  all  be  ellipses.  Hence  the 
surface  is  called  an  ellipsoid. 


If  we  suppose  z  =  ±  c,  we  have  x  =  0  and  ?/  =  0,  as  tlio 
only  values  of  x  and  y  which  can  satisfy  the  equation.  Hence 
each  of  the  two  planes  z  =  -\-  c  and  z  =  —  c  meets  the  sur- 
face in  a  single  point  on  the  axis  of  Z,  and  is  therefore  tan- 
gent to  the  surface.  Extending  the  same  proof  to  the  other 
two  co-ordinates,  we  reach  the  conclusion: 

The  e^kt  planes  x  =  -\-  a,  x  =  —a,  y  =  -{-h,  y  =  —  h, 
z  =  -{-  c  and  z  =  —  c  are  all  tajigeiits  to  the  ellipsoid  at  the 
points  which  lie  on  the  axes  at  the  distances  ±  a^  ±b  and  ±  c 
from  the  origin. 

These  six  planes  form  the  faces  of  a  rectangular  parallelo- 
piped  whose  edges  are  respectively  %a,  2b  and  2c.  Each  pair 
of  parallel  faces  being  at  equal  distances  on  the  two  sides  of 
the  origin,  and  parallel  to  the  corresponding  axes,  these  axes 
intersect  the  faces  in  their  centres.     Hence: 

Theorem  VIII.  Every  ellipsoid  may  be  inscribed  in  a 
rectajigular  parallelojjiped  whose  surface  it  will  touch  in  the 
centre  of  each  face. 

260.  Class  II.  The  Hyperboloid  of  One  Nappe.  Let  us 
take  that  form  of  the  equation  (7)  in  which  one  of  the  three 


QUADBIG  SURFACES.  285 

terms  of  the  first  member  is  negative.     Suppose  this  term  to 
be  that  in  z.     The  equation  is  then 

^V  ^:  _ !!  - 1  (8) 

which  we  may  write  in  the  form 

3  2  2 

l  +  F  =  ^  +  ?-  .         (^') 

Let  us  now  find  the  curve  in  which  the  surface  intersects 
the  plane  of  XY.  This  we  do  by  putting  2  =  0,  which  gives 
at  once  the  equation  of  an  ellipse  whose  major  axes  are  a  and 
h.     Hence: 

Theorem  IX.  The  hyperholoid  of  one  nappe  intersects 
the  plane  of  XY  in  an  ellipse  whose  axes  are  the  same  as  the 
axes'  a  and  I  of  the  surface. 

This  ellipse  is  called  the  ellipse  of  the  gorge. 

Let  us  next  find  the  curve  in  which  the  surface  intersects 
a  plane  parallel  to  that  of  XY  and  at  a  distance  k  from  it. 
The  equation  of  such  a  plane  is 

z  =  k. 
Substituting  this  constant  value  of  z,  and  putting,  for  brevity, 

V  =  l  +  %,  (a) 


c 


the  equation  (8')  reduces  to 


This  is  the  equation  of  an  ellipse  whose  axes  are  ha  and 
hh.  Whatever  the  value  of  h,  the  ratio  of  these  axes  will  be 
a  :  h,  so  that  the  ellipses  will  be  similar.     Hence: 

Theorem  X.  The  hyperloloid  of  one  nappe  cuts  all  planes 
perpendicular  to  its  axis  of  Z  in  similar  ellipses. 

The  equation  {a)  shows  that  h  exceeds  unity  and  increases 
with  positively  or  negatively  increasing  values  of  k.     Hence 

The  ellipses  in  which  the  hyperholoid  of  one  napjje  cuts 
planes  perpendicular  to  its  axis  of  Z  are  larger  the  farther  the 
planes  are  from  the  centre. 


286  OEOMETRY  OF  THREE  DIMENSIONS. 

To  find  the  curves  in  wliicli  the  surface  intersects  planes 
parallel  to  the  other  co-ordinate  planes,  we  transpose  either 
the  term  in  x  or  that  in  y,  thus  putting  the  equation  in  the 
form 

Assigning  any  constant  value  to  y,  we  see  that  the  equation  is 
that  of  an  hyperbola,  and  we  may  show,  as  in  the  case  of  the 
other  section,  that  these  hyperbolas  are  all  similar.  But 
there  is  one  remarkable  case,  namely,  that  in  which  the  equa- 
tion of  the  intersecting  plane  is 

y=±h. 

The  equation  of  the  intei'section  then  becomes 

a        c 
or  {ex  —  az)(cx  -j-  az)  =  0, 

which  is  the  equation  of  a  pair  of  straight  lines. 

This  result  will  be  generalized  hereafter. 

261.  Class  III.  Hyj^erholoid  of  Tiuo  Nappes.  Let  two 
of  the  terms  in  (6)  be  negative.  By  taking  the  terms  in  x 
and  y  as  negative,  and  then  changing  the  sign  of  each  mem- 
ber of  the  equation,  it  may  be  w^ritten  in  the  form 

If  c  >  2;  >  —  c,  the  second  member  will  be  negative  and 
the  equation  can  be  satisfied  by  no  real  values  of  x  and  y. 
When  z  is  on  either  side  of  the  limits  ±  c,  there  will  be  real 
values  of  x  and  y.  Hence  the  surface  is  composed  of  two  dis- 
tinct slieets,  or  naypes,  separated  at  their  nearest  points  hy  the 
sp)ace  2c.  This  surface  is  therefore  called  the  hyperboloid 
of  two  nappes. 

We  readily  see  that  the  j^lane  z  =  k,  parallel  to  the  plane 
of  XY,  intersects  the  surface  in  a  real  ellipse  whenever  k  >  c. 

We  also  show,  as  in  the  last  section,  that  the  planes  x  =  k 
and  y  =  h  intersect  it  in  hyperbolas. 


qUADRIG  SURFACES.  287 


Tangent    and    Polar  Lines  and  Planes  to   a 
Qnadric. 

362.  Since  the  equation  of  the  general  quadric  surface 
may  be  reduced  to  one  of  the  three  forms  just  considered,  we 
may,  without  loss  of  generality,  consider  the  equations  (6)  as 
representing  every  such  surface.  Moreover,  we  may,  in 
beginning,  restrict  ourselves  to  the  first  form, 

1  +  1+-?  =  !'  (11) 

because  the  fact  that  any  of  the  three  terms  of  the  first  mem- 
ber has  the  negative  sign  may  be  indicated  by  substituting 

—  a",  —  J*  or  —  c"         for        a",  ¥  or  c^, 

Def.  A  tangent  line  to  a  surface  is  a  line  which  passes 
through  two  coincident  points  of  the  surface. 

Pkoblem.  To  find  the  condition  that  a  line  shall  touch  a 
surface  of  the  second  order  at  thepoi?it  {x^,  y^,  z^)  on  that  sur- 
face. 

Solution.  Since  the  line  passes  through  the  point  {x^,  y^,  z^) 
of  tangency,  its  equations  may  be  written  in  the  form 

X  =  x^  +  Ip;    ^ 

y  =  yi  +  ^^p;  [  («) 

z  —  z^  -{-  nf>,  ) 

So  long  as  I,  m  and  n  are  unrestricted,  these  equations  may 
represent  any  line  through  the  point  (x^,  y^,  z^). 

To  find  the  points  in  which  the  line  meets  the  surface  (11), 
we  must  substitute  these  values  of  x,  y  and  z  in  the  equation 
of  the  surface.  Doing  this,  and  arranging  the  equations  in 
powers  of  p,  we  have  the  condition 

^r"  4_  y^"   ,    \"       1  4-  2/)f  ^^^  4-  ^y^  4-  ^^^^ 


288  GEOMETRY  OF  THREE  DIMENSIONS. 

Since  the  point  {x^,  ?/„  2,)  lies  on  the  surface  by  hypothe- 
sis, we  have 

from  which  it  follows  that  p  =  0  is  a  root  of  (b).  This  gives 
the  point  {x^,  y^,  z^  as  one  of  the  points,  as  it  ought  to. 
Dividing  by  p,  the  equation  becomes 

which  gives,  for  the  other  value  of  p, 

J  Ix.    .    mil.    .    nz\        r    .   vf   .    w'  .  ,. 

We  have  hitherto  subjected  the  line  {a)  to  no  restriction 
except  that  of  passing  through  the  point  {x^,  y^,  z^.  The 
equation  (^Z)  gives  the  value  of  p  in  terms  of  I,  m  and  n  for 
the  second  point  in  which  the  line  intersects  the  surface. 

Now,  the  problem  requires  that  this  second  point  shall 
coincide  with  the  first  one,  that  is,  that  p  =  0  in  {d).  This 
gives 

as  the  required  condition  that  the  line  A  shall  touch  the 
quadric  at  the  point  (x^,  y^,  z^). 

All  the  quantities  except  I,  m  and  n  in  this  equation  being 
regarded  as  given  constants,  it  constitutes  a  linear  equation 
between  I,  m  and  n.  Hence,  by  §  247  {h),  it  requires  that 
the  tangent  line  lie  in  the  plane 


X 


^,{x  -  x^)  +  -|(y  -  y.)  +  ii(2  -  z,)  =  0, 
which,  by  (c),  readily  reduces  to 

Hence  we  reach  the  conclusion: 


qUADRIC  SURFACES.  280 

Theorem  XL  All  straight  lines  toiccliing  a  quadric  sur- 
face at  the  same  point  lie  in  a  certain  plane  passing  through 
that  jioint. 

Def.  The  plane  containing  all  lines  tangent  to  a  surface 
at  the  same  point  is  called  a  tangent  plane  to  the  surface, 
and  is  said  to  touch  the  surface  at  that  jwint. 

263.  Lines  upon  the  Hyperloloid  of  One  Nappe.  The 
result  of  §260,  that  a  plane  may  intersect  an  hyperboloid  in  a 
pair  of  straight  lines,  is  a  special  case  of  the  following  theorem: 

Theorem  XII.  Through  every  point  upon  the  hyperloloid 
of  one  nappe  pass  tioo  straight  lines  which  lie  luholly  on  the 
surface,  and  ivhich  form  the  intersection  of  the  plane  tangent 
at  that  point  tvith  the  stcrface. 

Proof  We  may  write  the  equation  (9)  of  the  hyperboloid 
in  the  form 

(-:+3(-:-a=(^+f)(-i)-    (*) 

Now,  putting  A  for  an  arbitrary  constant,  let  us  consider 
the  two  planes  whose  equations  are: 

First  plane,     ^+ i=  ^(i  +  |); ") 

I       I      1  ,       \  (^) 

Second  plane, =  -Al  —  ^].   \ 

^  a       c      X\  h)    } 

Every  set  of  values  of  x,  y  and  z  which  satisfies  these  two 
equations  simultaneously  satisfies  the  equation  {h)  of  the  sur- 
face, as  we  readily  find  by  multiplication.  But  all  such 
values  belong  to  the  line  in  which  the  two  planes  intersect. 
Hence  this  line  lies  wholly  in  the  surface. 

We  have  next  to  show  that,  by  giving  a  suitable  value  to 
X,  this  line  may  pass  through  any  point  of  the  surface.  Let 
us  put  {x^,  ^j,  ^j),  the  point  through  which  the  line  is  to  pass. 

The  factor  A  must  then  satisfy  the  two  equations 


5a  +  .!.  =  A(l  +  f); 


a        c 


a       ~c        X\        bl' 


290  GEOMETRY  OF  THREE  DIMENSIONS. 

whence 

a        c  b  b        a        c  ^  ^ 

These  two  equations  give  the  same  value  of  X  when  a:,,  y^ 
and  z^  are  required  to  satisfy  the  equation  of  the  surface. 
Substituting  the  second  vahie  of  X  in  the  first  equation  (c)  of 
the  line  of  intersection,  and  the  first  value  in  the  second, 
these  equations  readily  reduce  to 

Taking  half  the  sum  and  half  the  difference  of  these  equations, 
they  become 


(") 


which  are  still  the  equations  of  the  line  in  question,  in  another 
form.  But  the  first  of  these  equations  is  that  of  the  tangent 
plane.  Hence  the  line  lies  in  the  tangent  plane  as  well  as  on 
the  surface,  and  therefore  forms  the  intersection  of  the  plane 
with  the  surface. 

The  other  line  through  {x^,  y^,  2,)  is  found,  in  the  same 
way,  to  be  given  by  the  simultaneous  equations 


^r^  ,  y.y 

a'  "^   b' 

0'  -  ^' 

^1^  ,  !/ 

ac~^  b 

ac        b 

X    .   z          L       y\ 

a       c       /x\        b  J 

The  value  of  ju,  found  like  that  of  A,  is 

a        c                0                0 

a 

z 
c 

Thus  the  equations  of  the  second  line  become  the  same  as 


qUADBIC  SURFACES. 


21J1 


those  already  found  for  the  first  one,  except  that  the  signs  of 
y^  and  y  are  changed.     In  part,  we  find 


x,x 

+ 

= 

1; 

ac 

— 

y 

x^z 
ac 

= 

— 

(/) 

which  are  the  equations  of  another  line  in  the  surface  and 
passing  through  (x^,  y^),  thus  proving  the  theorem. 

264,  The  equations  {e)  and  (/)  represent  two  lines, 
each  situated  both  in  the  surface  and  in  the  tangent  plane. 
Hence  the  theorem  may  be  expressed  in  the  form: 

Theorem  XIII.  Every  tangent  plane  to  the  hyperloloid 
of  one  nappe  intersects  the  surface  in  a  pair  of  straight  lines 
passing  through  the  point  of  tangency. 

It  is  evident  from  the  preceding  theory  that  the  surface  in  question 
may  be  generated  by  the  motion  of  a  line.  We  present  three  figures 
showing  the  relations  which  have  been  discussed.  In  the  first,  OP  is  a 
central  axis  or  rod,  supported  on  a  fixed  disk  at  the  bottom  and  carrying 


a  similar  disk  at  the  top.  The  latter  can  be  turned  round  on  the  rod. 
Vertical  threads  pass  from  all  points  of  the  circumference  of  one  disk  to 
the  corresponding  parts  of  the  other,  thus  forming  a  cylindrical  surface. 
Then  turning  the  upper  disk  through  any  angle  less  than  180°,  the 
threads  will  form  an  hyperboloid  of  revolution,  as  shown  in  the  other 


292  OEOMETRY  OF  THREE  DIMENSIONS. 

figure.  The  threads  shown  in  the  figure  are  those  of  one  system  only ; 
by  rotating  the  disk  in  the  opposite  direction  the  threads  would  be  those 
of  the  other  system. 

The  next  figure  represents  the  surface  as  cut  by  a  plane  very  near  the 
tangent  plane,  the  section  being  an  hyperbola  of  which  the  transverse 


axis  is  vertical.  By  moving  the  cutting  plane  a  little  closer  to  the 
centre,  the  bounding  curves  of  the  section  will  merge  into  the  dotted 
lines,  and  the  plane  will  be  a  tangent  to  the  surface  at  their  point  of 
intersection. 

265.  Equations  of  the  Generatitig  Lines.  To  study  the 
lines  in  question,  let  us  refer  each  to  the  point  in  which  it  in- 
tersects the  plane  of  XY.     We  shall  then  have  z^  =  0  and 

a'  ^   ¥ 
The  equations  of  any  one  of  the  first  set  of  lines  will  then 
become 

y       ^x^  _  III 
h        ac        h' 

Because  the  lines  lie  in  both  of  the  planes  represented  by  these 
equations  taken  singly,  the  coefficients  ?,  m  and  7i  in  the  vec- 
torial form  must,  by  §  347,  satisfy  the  conditions 


QUADBIC  SURFACES.  293 

!?'+>  =  '' 

0        ac 

These  equations  give  the  following  values  of  I  and  m  in 
terms  of  oi,  which  remains  arbitrary: 


7  ^^1 

bx, 
m  =       — -71. 
ac 


First  system  of  lines. 


Proceeding  in  the  same  way  with  the  second  set  of  lines, 
we  find,  starting  from  the  equations  (/), 

^  +%i  =0; 
b        ac 


from  which 


7  ^Vi 

m  = -n. 

ac 


Second  system  of  lines. 


If  we  give  n  in  both  systems  the  value  ale,  so  as  to 
avoid  fractions,  the  values  of  the  direction-vectors  I,  m  and 
n  will  be: 

/  I  =  -a'y,;  (  V  =+«>,; 

First  system:  ^  m  =  +  Vx^-,     Second  system:  -j  m'  =  —  Z'^o:,; 

( n  =        ahc'j  ( n'  =        ahc) 

the  values  of  the  second  system  being  distinguished  by  accents. 

266.  Theorem  XIV.  On  an  hyperholoid  every  line  of 
tlie  one  system  intersects  all  the  lines  of  the  other  system.  But 
no  tiuo  lines  of  the  same  system  intersect  each  other. 

Proof  Retaining  {x^,  y^,  0)  as  the  fundamental  point  of 
any  line  of  the  first  system,  and  putting  x^  and  y^  for  the 


294  GEOMETRY  OF  THREE  DIMENSIONS. 

values  of  ;r,  and  y^  in  case  of  any  line  of  the  second  system, 
the  condition  of  intersection  of  two  lines  (§  249)  will  be 

(mn'  —  7n'n){x^  —  x^)  +  {nV  -  n'l){y^  -  yj  =  0,       (a) 

the  third  term  being  omitted  because  z^  =  0  and  z^  =  0. 

If  we  substitute  for  m,  m',  etc.,  their  values,  as  just  given, 
this  equation  becomes 

ab'c{x^  +  x^)(x^  -  X,)  +  a'bc(y^  +  y,){y,  -  y,)  =  0. 

Dividing  by  a'^'c,  we  find  it  reduce  to 


a'  ^  V        \a'  ^  b'J 


Now,  by  hypothesis,  {x^,  y^  and  (x^,  y^  are  points  on  the 
ellipse  in  which  the  plane  of  XY  intersects  the  surface;  that 
is,  on  the  ellipse  whose  equation  is 

cC  ^  h' 

Hence  the  condition  reduces  to  1  —  1  =  0,  which  is  an  iden- 
tity, showing  that  the  lines  intersect. 

Secondly.  Let  both  lines  belong  to  the  same  system,  the 
one  line  intersecting  the  ellipse  of  the  plane  of  XY  in  the 
point  {x^,  y^),  as  before,  and  the  other  in  the  point  {x^,  y^). 
We  shall  then  have,  for  the  values  of  the  direction-vectors, 

m  =  +  b^x^;        m'  —  +  lfx^\ 
n  =  dbc\  n'  =  cibc. 

The  condition  {a)  of  intersection  then  becomes 

Each  term  being  a  perfect  square  is  necessarily  positive,  so 
that  the  condition  of  intersection  is  impossible.     Q.  E.  D. 

267.  Poles  and  Polar  Planes  of  the  Quadric.  Let  us 
consider  all  possible  tangent  planes  which  pass  through  a 
fixed  point  {x^,  y^,  z^)  not  belonging  to  the  quadric.  Let 
(^15  !/p  ^i)  ^6  t^6  variable  point  of  tangency  on  the  quadric. 


qUADRIC  SURFACES.  295 

The  equation  of  the  plane  tangent  at  (.Tj,  y^,  z^)  will  then  be 
(13).  The  condition  that  this  plane  shall  pass  through  the 
point  (x^,  y^,  z^)  is  that  the  co-ordinates  of  this  point  shall 
satisfy  the  equation  of  the  plane,  which  gives 

^>-o^M.  +  !4p_i^0.  (14) 


a"      '     b' 


This  is  now  a  condition  which  the  point  of  taugency  (a;,,  ?/„  z^) 
must  satisfy  as  it  varies.  Being  of  the  first  degree,  it  shows 
that  this  point  must  lie  in  a  certain  plane.  The  equation  of 
this  plane  may  be  written 

Def.  That  plane  which  contains  the  points  of  tangency 
of  all  tangents  to  a  quadric  which  pass  through  a  point  is 
called  the  polar  plane  of  that  point. 

The  point  is  called  the  pole  of  the  plane. 

Eemark  1.  The  point  of  tangency  in  the  aboye  case  may 
move  along  a  curve  which  ^\\\\  then  be  the  intersection  of 
the  polar  plane  and  the  quadric. 

Eemark  2.  The  point  {x^,  y^,  z^)  may  be  so  situated  that 
no  .real  tangent  plane  can  pass  through  it;  for  example,  in 
the  interior  of  an  ellipsoid.  The  points  of  tangency  (a;,,  y^,  z^) 
in  (14)  will  then  be  entirely  imaginary.  But  the  plane  (15) 
will  always  be  real  and  determinate;  only  it  will  not  meet  the 
quadric.     Hence: 

Theorem  XV.  To  every  point  in  space  correspo7ids  a 
defijiite  polar  plane  relative  to  any  quadric. 

Theorem  XVI.  Conversely,  To  every  plane  corresponds 
a  certain  pole. 

Proof,     Let  the  plane  be 

Ax  +  By  -^  Cz^  D  =  0,  {a) 

and  let  a,  h  and  c  be,  as  before,  the  principal  axes  of  the 
quadric.  Comparing  the  equation  {a)  with  (15),  we  see  that 
they  become  identical  if  we  can  have 

^o__^.  .lo__^.  'io__C 

fi^  ~       /)'         h''~       />'         o'  ~       D 


296  GEOMETRY  OF  THREE  DIMENSIONS. 

This  only  requires  that  we  determine  x^,  y^  and  z^  by  the 
conditions 

__«M,  __^.  -_£!^. 

which  always  give  real  values  of  x^,  y^  and  z^,  and  therefore  a 
real  pole.     Q.  E.  D. 

Cor.  If  the  plane  approach  the  centre  as  a  limit,  D  ap- 
proaches zero  as  its  limit,  and  x^,  y^  and  z^  increase  indefi- 
nitely.    Hence 

The  pole  of  any  diametral  plaJie  of  a  quadric  is  at  infinity. 

Notation.  If  we  call  any  points  P,  Q,  etc.,  we  shall  call 
their  polar  planes  P',  Q',  etc. 

268.  Theorem  XVII.  If  a  point  lie  onaplane,  thepole 
of  the  plane  ivill  lie  07i  the  polar  plane  of  the  point. 

Proof  Let  a  point  P  be  {x^,  y^,  z^)  and  a  point  Q  be 
(x^,  y^,  z^).     Then,  by  (15),  the  polar  plane  §'  is 

a^     ^     If     ^    c- 

Let  the  point  P  lie  on  this  plane.  Then  the  co-ordinates 
x^,  y^,  z^  must  satisfy  this  equation;  that  is, 

a'    '^    V"'^  &  ' 

This  equation  shows  that  the  co-ordinates  (x^^  y^^  z^  satisfy 
the  equation 

which  is  the  equation  of  the  polar  plane  of  {x^,  y^,  z^).  Hence 
the  pole  {x^,  y^,  z^y  or  Q,  lies  on  this  plane.     Q.  E.  D. 

Cor.  If  any  nunriber  of  points  lie  in  a  plane  P',  iheir 
polar  planes  loill  all  pass  through  the  pole  P  of  that  plane. 

Conversely,  If  any  number  of  planes  pass  through  a  point 
Q,  their  poles  will  all  lie  on  the  polar  plane  of  Q. 

Theorem  XVIII.  If  any  member  of  planes  iiitersect  in 
a  straight  line,  their  poles  will  all  lie  i7i  another  straight  line. 


QUADRIC  SURFACES.  297 

Proof.  In  order  that  planes  may  intersect  in  one  line,  it 
is  necessary  and  sufficient  that  they  should  all  pass  through 
any  two  points,  taken  at  pleasure,  on  that  line.  Let  P  and 
Q  be  two  such  points.     Then — 

Because  all  the  polar  planes  pass  through  the  point  P, 
their  poles  all  lie  somewhere  in  the  polar  plane  P'; 

Because  these  planes  all  pass  through  the  point  Q,  their 
poles  all  lie  somewhere  in  the  polar  plane  Q'. 

Hence  these  poles  all  lie  on  the  intersection  of  P' and  Q', 
which  is  a  straiglit  line.     Q.  E.  D. 

Cor.  It  is  readily  shown  by  reversing  the  course  of  reason- 
ing that  if  any  numher  of  points  lie  in  a  straight  line,  their 
jjolar  planes  ivill  all  pass  through  another  line. 

Def  Two  lines  so  related  that  all  poles  of  planes  passing 
through  one  lie  in  the  other  are  called  reciprocal  polars. 


EXERCISES 

1.  If  an  ellipsoid,  an  hyperboloid  of  one  nappe  and  one  of 
two  nappes  are  formed  with  the  same  principal  axes,  a,  h,  c, 
it  is  required  to  write  the  equations  of  their  several  polar 
planes  relatively  to  the  pole  {x^,  y^,  zj. 

2.  In  this  case  show  that  the  polar  planes  with  respect  to 
the  ellipsoid  and  the  hyperboloid  of  one  nappe  intersect  in  the 
plane  of  XT  on  the  line  b'^x^x  -\-  a^y^y  —  1  =  0. 

3.  In  the  same  case  show  that  the  polar  planes  with  respect 
to  the  hyperboloids  of  one  and  of  two  nappes  respectively  are 
parallel. 

4.  In  the  same  case  show  that  the  polar  planes  with  respect 
to  the  ellipse  and  the  hyperboloid  of  two  nappes  respectively 
intersect  in  a  line  parallel  to  the  plane  of  XY  and  intersecting 
the  axis  of  Z. 

5.  Show  that  if  a  pole  lies  on  any  diameter,  the  polar  plane 
will  be  parallel  to  the  diametral  plane  conjugate  to  such  dia- 
meter. 

6.  Show  that  if  the  pole  lie  on  the  surface  of  the  quadric, 
the  polar  plane  will  touch  the  surface  at  the  pole. 


298  GEOMETRY  OF  TUREE  DIMENSIONS. 

1.  Show  that  the  reciprocal  polar  of  a  line  tangent  to  a 
quadric  is  another  line  tangent  at  the  same  point,  the  two 
tangents  lying  in  a  pair  of  conjugate  diametral  planes. 

8.  If  a  line  is  required  to  lie  in  a  diametral  plane,  show 
that  its  reciprocal  polar  must  be  parallel  to  the  diameter  con- 
jugate to  that  plane. 

Special  Cases  of  Quaclrics. 

269.  In  all  the  preceding  investigations  it  has  been 
assumed  that  the  co-ordinates  A,  B,  C  of  the  centre,  given  by 
the  equations  (2),  and  the  quantities  I',  m',  w'  and  d'  in  (5), 
which  determine  the  three  principal  axes  of  the  quadric,  are 
sM  finite  and  deter mmate. 

Although  in  the  general  case  this  will  be  true,  yet  the  nine 
constants  which  determine  the  quadric  may  have  such  special 
values  that  these  quantities  may  be  zero  or  infinite.  The 
complete  discussion  of  these  cases  would  require  us  to  make 
extensive  use  of  the  theory  of  determinant^,  which  we  wish 
to  avoid;  we  therefore  shall  merely  point  out  to  the  student 
the  possibility  of  certain  special  cases. 

370.  The  Paraboloid.  When  we  solve  a  system  of  three 
equations  with  three  unknown  quantities,  like  (2),  each  un- 
known quantity  comes  out  as  the  quotient  of  two  quantities 
(compare  eq.  (3)  of  §  188  for  example),  and  the  denominator 
of  these  quotients  is  the  same  for  all  the  quantities.  If  this 
denominator  approaches  zero  as  a  limit,  the  values  of  A,  B 
and  C  in  (2)  will  increase  without  limit.  Hence,  if  this  de- 
nominator vanishes,  the  centre  of  the  quadric  is  at  infinity. 

In  this  case  the  quadric  is  called  a  paraboloid. 

271.  The  Cone.  In  reducing  the  original  equation  to 
the  form  (3),  the  absolute  term  d'  may  vanish.  In  this  case 
the  principal  axes  a,  h  and  c  (§  257)  will  all  vanish  (unless 
some  of  the  quantities  V,  m'  or  n'  are  also  zero),  and  we  sliall 
have  the  homogeneous  equation 

(jx^  -h  hf  +  hz""  +  %(}'yz  +  2AV'c  +  'Ih'xy  =  0.       (16) 

Def.     A  cone  is  a  surface  generated  by  the  motion  of  a 


QUADIUC  SURFACES.  299 

line  which  passes  through  a  fixed  point  and  continually  in- 
tersects a  fixed  curve. 

The  fixed  point  is  called  tlie  vertex  of  the  cone. 

The  fixed  curve  is  called  the  directrix  of  the  cone. 

A  quadric  cone  is  one  whose  directrix  is  a  plane  locus 
of  the  second  degree. 

Theorem  XIX.  Every  liomogeneous  equation  of  the 
second  degree  has  for  its  locus  a  quadric  cone  lohose  vertex  is 
at  the  origin. 

Taking  the  equation  (16),  which  is  a  perfectly  general  one 
of  the  kind  named  in  the  theorem,  we  first  prove  that  its  locus 
is  a  cone  having  its  vertex  at  the  origin,  in  the  following  way: 

We  take  any  point  at  pleasure  on  the  surface  (IG); 

"We  pass  a  line  through  this  point  and  through  the  origin; 

We  then  show  that  this  line  must  lie  wholly  on  the  locus. 

Let  {x^,  y^y  z^)  be  any  point  on  the  surface  (16).  Then 
every  point  {Xy  y,  z)  determined  by  the  equations 

X  =  px^,  J 

y  =  PVv  [    ■     .  («) 

z  =  pz^, ) 

will  lie  on  the  line  passing  through  the  origin  and  (a;,,  y^,  z^). 
Substituting  these  values  in  (16)  gives,  for  the  condition  that 
the  point  {x,  y,  z)  shall  lie  on  the  locus, 

By  hypothesis  {x^,  y^,  z^)  satisfies  (16).  Hence  this  condition 
{h)  is  satisfied  for  all  values  of  p;  hence  every  point  deter- 
mined by  (a)  lies  on  the  surface  (16);  Avhence  this  surface  is 
so77ie  cone. 

Secondly,  being  of  the  second  degree,  (16)  represents  a 
quadric  surface;  whence,  by  Th.  IV.,  every  plane  intersects  it 
in  a  conic,  and  it  is  by  definition  a  quadric  cone. 

Kemark.  For  the  directrix  of  the  cone  we  may  take  its 
intersection  with  any  plane  whatever  not  passing  through  the 
vertex.  Let  us  then  take  the  plane  z  =  c.  We  then  shall 
have  from  (16),  for  the  equation  of  the  directrix, 

gx'  _|_  ky''  +  2k'xy  +  2k'cx  +  2g'cy  +  kc'  =  0.         (17) 


300  GEOMETRY  OF  THREE  DIMENSIONS. 

The  coefficients  being  all  independent,  this  curve  may  be  any 
conic  whatever.  Hence  (16)  may  represent  any  quadric  cone 
whose  vertex  is  at  the  origin. 

2*72.  Special  Case  lolien  a  Quadric  becomes  a  Pair  of 
Planes.  Since  the  directrix  of  the  cone  may  be  any  conic,  it 
may  be  a  pair  of  straight  lines.  Since  a  line  turning  on  a 
point  and  intersecting  a  fixed  line  describes  a  plane,  it  fol- 
lows that  lohenever  the  directrix  is  a  pair  of  lines,  the  quadric 
cone  becomes  a  pair  of  planes.  Hence  among  the  special  kinds 
of  quadrics  must  be  included  a  pair  of  planes. 

The  quadric  equation  of  a  given  pair  oi  planes  is  readily 
found.     If  the  equations  of  the  planes  are 

ax  -\-  by   -{-  cz   -\-  d   =0, 
a'x  +  b'y  +  c'z  -f  d'  =  0, 

we  liave  only  to  take  the  product  of  these  equations,  which 
will  be  of  the  second  degree  in  x,  y  and  z. 

273.  Surfaces  of  Revolution.  In  the  reduction  of  the 
general  quadric,  two  of  the  principal  axes,  a  and  J  for  example, 
may  be  found  equal.  In  this  case  the  equation  may  be  re- 
duced to  one  of  the  forms 


a  c 

or 


a  c' 

x'  +  f±a'[^,±lj==0.  (17) 


Assigning  any  constant  value  to  z,  the  equation  in  x  and  y 
will  be  that  of  a  circle.  Hence  all  planes  parallel  to  the  plane 
of  XFwill  intersect  the  quadric  in  circles  having  their  centres 
on  the  axis  of  Z.  Since  all  sections  containing  the  axis  of  Z 
will  be  conies,  the  surface  can  be  generated  by  the  revolution 
of  some  conic  around  the  axis  of  Z.  It  is  therefore  called  a 
surface  of  revolution. 

The  equation  (17)  admits  of  the  same  four-fold  classifica- 
tion as  the  equation  (6),  according  to  the  algebraic  signs  of 
the  ambiguous  terms.  We  have  therefore,  as  the  three  real 
forms — 


QUADRIG  SURFACES.  301 

I.  The  ellipsoid  of  revohdioji : 

II.  Tlie  hyperboloid  of  revolution  of  one  nappe: 

III.  The  hyperboloid  of  revolution  of  two  nappes: 

When,  in  the  ellipsoid, 

c  >  a,  the  ellipsoid  is  called  prolate ; 
c  <  «,  the  ellipsoid  is  called  oblate ; 
c  —  a,  the  ellipsoid  is  called  a  sphere. 

In  the  hyperboloid  of  one  nappe  the  axis  c  may  be  infi- 
nite.    The  equation  will  then  be 

x^  ^  if  ^  a\ 

the  equation  of  a  cylinder  of  radius  «,  whose  axis  is  that  ofZ. 

274.  Deriving  Surf  aces  from  the  Generating  Curve.  The 
general  method  by  which  we  find  the  equation  of  the  surface 
generated  by  revolving  a  curve  around  the  axis  of  Z  is  this: 

Assume  the  curve  to  be  in  any  initial  position. 

Take  any  point  upon  it  whose  vertical  ordinate  is  z,  and 
find  the  corresponding  values  of  x  and  y,  and  hence  of 
Vx"  +  ?/",  in  terms  of  z,  the  distance  of  the  point  from  the 
plane  of  XY. 

Since  this  distance  remains  constant  while  the  point  re- 
volves, the  square  of  the  equation  thus  found  will  be  the 
equation  of  the  surface. 

If  the  fixed  position  can  be  so  chosen  that  the  generating 
curve  may  lie  wholly  in  the  plane  of  XZ  (or  YZ),  one  of  the 
co-ordinates  y  or  z  will  then  be  zero,  and  we  have  only  to  sub- 
stitute Vx"  +  if  for  X  or  y,  as  the  case  may  be. 


302  GEOMETRY  OF  THREE  DIMENSIONS. 

275,  The  Paraholoid  of  Revolution.  Let  us  suppose  a 
parabola  to  revolve  about  its  principal  axis,  wliicli  we  shall 
take  as  the  axis  of  Z.  The  square  of  each  ordinate  will  then 
be  2pz,  But  this  square,  as  the  curve  revolves,  is  continually 
equal  to  x"  -\-  y',  because  the  ordinate  is  a  line  perpendicular 
to  the  axis  of  Z,  whose  terminus  on  the  curve  is  represented 
by  the  co-ordinates  x  and  y  of  the  curve.  Hence  the  equa- 
tion of  the  surface  is 

x'  -\-y-  =  2pz', 
2)  being  the  semi-parameter  of  the  generating  parabola. 

EXERCISES. 

1.  Find  the  equation  of  the  cone  generated  by  revolving 
around  the  axis  of  Z  the  straight  line  whose  equation,  when 
tRe  line  is  in  the  plane  XZ,  is 

X  =  mz  -f  i- 
Find  also  the  vertex  of  the  cone. 

Aiis.  x^  -\-^f=:  m'z'  +  2mbz  +  b\ 

Vertex  at  the  point,  (o,  0, ). 

2.  Investigate  the  surface  generated  by  the  motion  of  a 
straight  line  around  an  axis  which  does  not  intersect  it,  the 
shortest  distance  of  the  line  from  the  axis  being  a,  and  the 
angle  between  them  being  a.  (In  the  initial  position  we  may 
suppose  the  line  to  intersect  the  axis  of  JT  at  right  angles  at 
the  distance  a  from  the  origin,  and  to  form  an  angle  a,  whose 
tangent  is  m,  with  the  axis  of  Z.)  Find  the  equation  of  the 
surface  and  its  principal  axes. 

A71S.  x^  -j-  y^—  m'^z'  =  a^.     Axes:  a,  a,  —. 

Here,  if  we  take  a  point  on  the  line  at  the  distance  r  from  the  axis 
of  X,  its  co-ordinates  in  the  initial  position  will  be 

x  =  a;      2  =  7*coscr;      y  =  ?'sin  a  =  2tan  a  =  ws. 

3.  Find  the  equation  of  the  cone  generated  by  the  revolu- 
tion of  the  straight  line  whose  equations  in  one  position  are 

x  =  mp;         y  =  0]         z  =  np. 

Ans.  n\x'  +  y')  -  m'z'  =  0. 


qUADRIC  SURFACES.  303 

4.  Find  the  equation  of  the  ellipsoid  generated  by  the  re- 
Tolution  of  the  ellipse  b'^x^  -\-  a^z^  =  a^b^, 

6.  If  the  hyperbola  cV  —  a^z'  =  aV  revolve  about  the 
axis  of  Z,  find  the  equation  of  the  curve  and  of  the  cone  de- 
scribed by  the  asymptotes. 

a  c 

a'  e  ~ 

6.  Investigate  the  surface  when  the  revolving  hyperbola 
is  a^z"  —  c'x^  —  a^y^. 

7.  Find  the  equation  of  the  surface  generated  by  the  revo- 
lution about  the  axis  of  Z  of  the  line  whose  equations  are 

X  =  a  -\-  lft\ 
y  z=l)  J^mp', 
z  =■  c  -\-  np. 

Ans.  n\x^  +  y')  =  {na  -  Uy  +  {nh  -  mcY  +  (P  +  m')z' 

-\-  2(nla  -\-  mnh  —  Pc  —■  m^c)z. 

8.  Show  that  the  equation  of  a  sphere  of  radius  r  whose 
centre  is  at  the  point  (a,  h,  c)  is 

(X  -  ay  +  {^J-  by  -\-{z-  cy  -  r'  =  0, 

and  find  the  value  of  r  in  order  that  the  origin  may  bisect  the 
radius  passing  through  it. 

9.  Find  the  plane  of  the  circle  in  which  the  spheres 

(x-ay-^iy-by-\-{z-cy  =r' 
and  (x  -  ay+  {y  -  by^  {z  -  cy  =  r" 

intersect  each  other. 

10.  Show  that  if  three  spheres  mutually  intersect  each 
other,  the  planes  of  their  three  circles  of  intersection  pass 
through  a  line  perpendicular  to  the  plane  containing  the 
centres.     (One  of  the  centres  may  be  taken  as  the  origin. ) 

11.  Investigate  the  locus  of  the  equation 

a  and  b  being  both  positive. 


304  GEOMETRY  OF  THREE  DIMENSIONS. 

12.  Do  the  same  thing  for  the  equation 

a         b 

13.  Show  that  the  six  planes  in  which  four  circles  taken 
two  and  two  intersect  each  other  all  pass  through  a  point. 

14.  Investigate  the  relations  of  the  three  surfaces 


a'  ^  b' 

-5  =  '. 

-^:=« 

-^-'' 

and 

Show  that  if  these  surfaces  be  cut  by  a  plane  parallel  to 
that  of  XY,  the  two  areas  included  between  the  three  ellipses 
of  intersection  will  each  be  constant  and  equal  to  the  area  of 
the  ellipse  in  which  the  first  surface  intersects  the  plane  of 
XF. 

15.  A  straight  line  moves  so  that  three  fixed  points  upon  it 
constantly  lie  in  the  three  co-ordinate  planes.  Find  the  locus 
of  a  fourth  point  upon  it  whose  distances  from  the  other  three 
points  are  a,  b  and  c. 

16.  From  the  results  of  §266  deduce  the  following  con- 
clusions: 

I.  The  cosine  of  the  angle  between  the  two  generating 
lines  through  the  point  (.r^,  y^,  0)  of  the  surface  is 

a^b\'  -  a'y^  -  b\^ 
a'b'c'  +  ay;  +  ^W 

II.  At  the  ends  of  the  respective  axes  a  and  b  the  cosines 
are    -^—. — tv    and 


c'  +  l^     ""^     c»  +  a'' 
III.  If  a  =  6  =  c,  the  lines  are  at  right  angles  to  each 
other. 


PART    III. 


INTRODUCTION   TO  MODERN 
GEOMETRY, 


276,  The  Principle  of  Duality.  In  modern  geometry 
every  straight  line  is  supposed  to  extend  out  indefinitely  in 
both  directions,  and  is  called  a  line  simi)ly.  Hence  lines,  like 
points,  differ  only  in  situation. 

Def.  A  segment  is  that  portion  of  a  line  contained  be- 
tween two  fixed  points.  Hence  a  segment  is  what  is  called  a 
finite  straight  line  in  elementary  geometry. 

There  are  certain  propositions  relating  to  lines  and  points 
which  remain  true  when  we  interchange  the  words  jyoiVi^  and 
line,  provided  that  we  suitably  interpret  the  connecting  words. 

The  principle  in  virtue  of  which  this  is  true  is  called  the 
principle  of  duality. 

Two  propositions  which  differ  only  in  that  the  words  point 
and  line  are  interchanged  are  said  to  be  correlative  to  each 
other. 

The  following  are  examples  of  correlative  propositions  and 
definitions.  The  right-hand  column  contains  in  each  case 
the  correlative  of  the  proposition  found  at  its  left. 

I.  Prop.  Through  any  I.  P7'op.  On  any  line 
point  may  pass  an  indefinite  may  lie  an  indefinite  number 
number  of  liries.                           oi  points. 

II.  Def.  Any  number  of  II.  Def.  Any  number  of 
lines  passing  through  a  point  points  lying  on  a  line  is  called 
is  called  a  pencil  of  lines,  a  row  of  points,  or  simply 
or  simply  a  pencil.                    a  point-row  or  row. 


306 


MODERN  GEOMETRY. 


The  common  point  of  a 
pencil  is  called  the  vertex 
of  the  pencil. 

III.  Prop.  Two  points 
determine  the  position  of  a 
certain  line,  namely,  the  line 
joining  them. 

IV.  Def.  The  line  join- 
ing two  points  is  called  the 
junction-line  of  the  points. 

V.  Prop.  Three  points, 
taken  two  and  two,  determine 
by  their  junction-lines  three 
lines. 

VI.  Prop.  A  collection 
of  n  lines,  taken  two  and  two, 

has,    m    general,    -^^ — 

junction-points. 

VII.  Def.  A  collection  of 
four  lines,  with  their  six  junc- 
tion-points, is  called  a  com- 
plete quadrilateral. 


The  line  on  which  a  row 
of  points  lie  is  called  the  car- 
rier of  the  row. 

III.  Prop.  Two  lines  de- 
termine the  position  of  a  cer- 
tain point,  namely,  their  point 
of  intersection. 

IV.  Def.  The  point  of  in- 
tersection of  two  lines  is  called 
the  junction-point  of  the 
lines. 

V.  Prop.  Three  lines, 
taken  two  and  two,  determine 
by  their  intersections  three 
junction-points. 

VI.  Prop.  A  collection 
of  n  points,  taken  two  and 

two,  has,  in  general,  -^^-^r — - 

K) 

junction-lines. 

VII.  Def.  A  collection 
of  four  points,  with  their  six 
junction-lines,  is  called  a 
complete  quadrangle. 


^^""Z 

9 

y^ 

2 
4 

VIII.  Prop.  On  each  of 
the  four  sides  of  a  complete 
quadrilateral  lie  three  ver- 
tices. 


VIII.  Prop.  Through 
each  of  the  four  vertices  of  a 
complete  quadi'angle 
three  sides. 


PRINCIPLE  OF  DUALITY. 


307 


IX.  Prop.  The  complete 
quadrilateral  has  three  diago- 
nals, formed  by  joining  the 
junction-point  of  each  two 
sides  to  the  junction-point  of 
the  remaining  two  sides. 

If  two  lines  are  represented 
by  the  symbols  a  and  h,  their 
junction-point  is  represented 
by  ah. 

The  pencil  of  lines  from  a 
vertex  P  to  the  points  a,  h,  c, 
etc.,  is  represented  by  P-ahc, 
etc. 

When  two  lines  are  each 
represented  by  a  pair  of  point- 
symbols,  a  comma  may  be  in- 
serted between  the  pairs  when 
their  junction-point  is  ex- 
pressed. 

Ex.  The  expression  ah,  xy 
means  the  junction-point  of 
the  lines  ah  and  xy. 


IX.  Prop.  The  complete 
quadrangle  has  three  minor 
vertices,  being  the  intersection 
of  the  junction-line  of  each 
two  vertices  with  the  junction- 
line  of  the  remaining  two. 

If  two  points  are  repre- 
sented by  the  symbols  a  and 
h,  their  junction-line  is  repre- 
sented by  ah. 

The  row  of  points  in  which 
a  carrier  R  intersects  the  lines 
A,  B,  G,  etc.,  is  represented 
by  R-ABC,  etc. 

When  two  points  are  each 
represented  by  a  pair  of  line- 
symbols,  a  comma  may  be  in- 
serted between  the  pairs  when 
their  junction-line  is  ex- 
pressed. 

Ex.  The  expression  AB, 
XY  means  the  junction-line 
of  the  points  AB  and  XY. 


Scholium.  When,  in  elementary  geometry,  two  intersecting  lines 
are  drawn,  their  junction-point,  being  evident  to  the  eye,  is  not  sepa- 
rately marked.  But  when  two  points  are  given,  it  is  considered  neces- 
sary to  draw  their  junction-line  wherever  this  line  is  referred  to.  But 
this  is  not  always  necessary  in  the  higher  geometry,  and  such  lines 
may  be  omitted  when  drawing  them  would  make  the  figure  too  compli- 
cated. 


308  MODERN  GEOMETRY. 


The  Distance-Ratio  and  its  Correlative. 

377.  The  Distance- Ratio.  Heretofore  the  position  of  a 
point  on  a  straight  line  has  been  expressed  by  its  distance 
(positive  or  negative)  from  some  other  point,  supposed  fixed 
on  the  line. 

The  position  of  the  point  may  also  be  expressed  by  the 
'ratio  of  its  distances  from  two  fixed  points  on  the  line. 


Let  a  and  h  be  two  fixed  points  on  an  indefinite  line, 
which  points  we  may  regard  as  the  ends  of  a  segment  ai  of 
the  line.  Let  x^,  x^  and  x^  be  three  positions  of  a  movable 
point  X,  and  let  us  consider  the  ratio  ax  :  hx  of  the  distances 
of  X  from  the  points  a  and  b.     If  we  put 

h,  the  distance  ab; 
h,  the  distance  ax\ 
r,  the  ratio  ax  :  bx. 


we  shall  have 


K 


(1) 


This  fraction,  or  ax  :  bx,  is  called  the  distance -ratio 
of  the  point  x  "with  respect  to  the  points  a  and  h. 

NoTATioiq^.     The  distance-ratio  is  written 

.  _ax 
(a.b,x)^-^-. 

Let  us  now  study  the  changes  of  value  of  the  distance- 
ratio  as  the  point  moves  along  the  line. 

Assuming  the  positive  direction  to  be  toward  the  right, 
then,  when  x  is  in  the  position  x^,  the  distances  ax^  and  bx^ 
will  both  be  positive,  and  we  shall  have 

r>+l.f  ^' 


THE  DISTANCE-RATIO.  305^ 

If  X  recedes  indefinifcely  toward  the  right,  k  increases  in- 
definitely and  the  ratio  t — y  appronclies   unity  as   its  limit. 

Tlierefore,  for  a  point  at  infinity  on  the  line,  we  have 

r  =  +  1. 
Supposing  the  point  to  move  toward  the  left,  the  denomi- 
nator k  —  li  will  become  zero  when  x  reaches  h;  and  as  tliis 
point  is  approached,  the  fraction  r  will  increase  without  limit. 
Hence,  when  x  is  at  Z>, 

r   =    00. 

When  X  is  in  the  position  x^  between  a  and  h,  ax  will  be 
positive  and  Ix  negative.     Hence,  in  this  part  of  the  line, 
r  =  a  negative  quantity. 
As  X  passes  from  h  to  a,  r  will  increase  from  negative  in- 
finity to  zero. 
At  a, 

r  =  0. 

In  the  position  x^  both  terms  of  the  ratio  will  be  negative, 
and  ?•  will  be  a  positive  proper  fraction. 

As  X  recedes  to  infinity  on  the  left,  r  will  approach  unity 
as  its  limit.  Hence,  whether  we  suppose  x  to  reach  infinity 
ill  the  negative  or  positive  direction,  we  have,  at  infinity, 

r  =  +  1, 

and  no  distinction  is  necessary  between  the  two  infinities. 

If,  then,  we  suppose  the  point  .t  to  move  along  the  whole 
line  from  negative  to  positive  infinity,  we  may  consider  it  as 
arriving  back  at  its  starting-point,  and  being  ready  to  repeat 
the  motion.  During  this  motion  the  distance-ratio  r  will 
also  have  gone  through  all  possible  values  from  negative  to 
positive  infinity,  and  will  be  back  at  its  starting-point.  The 
order  of  positions  of  the  jooint  and  the  order  of  changes  of  r 
are  as  follows: 

Point:    Infinity;  negative;  at  point  a\  on  fine  a&;  at  point  h;  posi- 
tive; infinity. 
Dist.  r.:  Unity;  positive  <  1;  zero;  negative;  infinity;  positive  >  1; 
unity. 


310 


MODERN  GEOMETRY. 


278.  To  exhibit  to  the  eye  the  changes  in  r  as  a;  moves 
along  the  hne,  we  may  erect  at  each  point  of  the  line  an  ordi- 
nate the  length  of  which  shall  represent  the  value  of  r  at 
that  point.  The  curve  passing  through  the  ends  of  the  ordi- 
nates  will  be  that  required. 


The  ratio  r  being  a  pure  number,  the  length  which  shall 
represent  unity  may  be  taken  at  pleasure.  So  we  may  lay 
down  from  the  middle  point  m  of  ab  an  arbitrary  length 
772^  =  —  1,  and  the  lengths  of  all  the  other  ordinates  will  be 
fixed. 

2*79.  Theorem.  T7ie  position  of  a  point  is  completely 
fixed  hy  its  distance-ratio  with  r'espect  to  tzvo  given  points; 
that  is,  there  can  he  only  one  point  on  the  line  to  correspond  to 
a  given  value  of  the  distance-ratio. 

This  is  the  same  as  saying  that,  in  the  equation  (1),  only 
one  value  of  k  will  correspond  to  given  values  of  r  and  h. 
This  is  readily  proved  by  solving  the  equation  with  respect  to 
h.  We  note  that  in  the  special  case  when  r  =  1  the  point  is 
at  infinity. 

280.  Relation  of  the  Distance- Ratio  to  the  Division  of  a 
Line.  The  conception  of  the  distance-ratio  occurs  in  elemen- 
tary geometry  when  we  say  that  the  point  x  divides  the  line 
al  internally  or  externally  into  the  segments  ax  and  hx,  having 


THE  SINE-RATIO.  311 

a  certain  ratio  to  each  other.  This  ratio  is  identical  with  the 
distance-ratio  just  defined.  It  is  negative  when  the  line  ab  is 
cut  internally;  positive  when  it  is  cut  externally. 

We  may  therefore,  instead  of  saying,  '^  The  distance-ratio 
of  the  point  x  with  respect  to  the  points  a  and  h,"  say, 
'*  The  ratio  in  which  the  point  x  divides  the  segment  ah.*^ 

EXERCISES. 

1.  Show  that  the  curve  which  expresses  the  value  of  r  in 
the  preceding  section  is  an  hyperbola;  find  its  asymptotes; 
define  the  class  to  which  it  belongs;  construct  its  major  axis 
in  the  case  when  we  take  mq  =  ab. 

2.  Show  that  if  we  take  two  points  at  equal  distances  on 
each  side  of  the  middle  point  m  of  the  base-line,  the  pro- 
duct of  the  corresponding  values  of  r  will  be  unity.  Trans- 
late this  result  into  a  property  of  the  equilateral  hyperbola. 

281.  The  Sine- Eatio,  In  the  two  preceding  articles  we 
showed  how  to  express  the  position  of  a  varying  jt?o^n^  upon  a 
fixed  line.  The  correlative  of  this  problem  is  that  of  express- 
ing the  position  of  a  varying  line  which  must  pass  through  a 
fixed  poi7it. 

As,  in  the  first  case,  the  position  of  the  moving  point  is 
expressed  by  its  relation  to  two  fixed  points  on  the  line,  so, 
in  the  second  case,  the  position  of  the  moving  line  is  fixed  by 
its  relation  to  two  fixed  lines  passing  through  the  point.  Let 
us  put  ^,  X 

0,  the  fixed  point;  i  5^ 

OA,    OB,    the  two    fixed  |       /  /  \ 

lines;  Bl.^..,^^^  1  /       'O^-^^ 

OX,  the  moving  line.  X^^^^^^^I^^^^j^-^^^^^' 

From  any  point  P  of  this    j^^-^"^'^/^^^     P^^^-^~-_ 
line  drop  the  perpendiculars  / 

FF'  and  FF''  upon  the  fixed  / 

lines.     Let  us  then  consider  the  ratio 

FP' 


pp,r 


312  MODERN  GEOMETRY. 

We  readily  see  that  the  value  of  r  is  the  same  in  whatever 
position  on  the  line  OX  the  point  P  is  chosen,  and  that 

sin  A  OX 


R  = 


sill  BOX' 


Hence  we  call  E  the  sine-ratio  of  the  line  OX  tvith  re- 
spect to  the  lines  OA  and  OB. 

To  investigate  the  algebraic  signs  of  sin  A  OXand  sin  ^  OX, 
let  us  take  the  directions  OA,  OB  and  OP  as  positive.  Then, 
in  accordance  with  the  usual  trigonometric  convention,  the 
sine  of  A  OP  will  be  positive  or  negative  according  as  a  person 
standing  at  0  and  facing  toward  A  has  the  point  P  on  the 
left  or  right  side  of  the  line  OA. 

Suppose  the  line  OX  to  start  from  the  position  OA  and  to 
turn  round  0  in  the  positive  direction.     Then, 

As  OX  starts  from  OA, 

R  starts  positively  from  zero. 
When  OX  reaches  the  bisector  OX', 

because  then  AOX  -{-  BOX  =  180°. 
As  OX  approaches  the  position  OB', 

R  increases  indefinitely, 

because  sin  ^  OX  approaches  zero. 
When  OX  reaches  OB', 

R  =  oo. 
As  OX  passes  from  OB  to  the  bisector  OX", 

R  increases  from  —  oo  to  —  1. 
When  OX  reaches  OX", 

R=  -1. 
As  OX  passes  from  OX"  to  OA', 

R  increases  from  —  1  to  0. 

The  line  OX  has  now  reached  its  initial  position,  though 
its  positive  direction  is  reversed.     Completing  the  revolution. 


DISTANCE-  AND  SINE-RATIO.  313 

we  see  tluit  R  goes  through  the  same  scries  of  vahies  as  before. 
Hence 

The  sine-ratio  depends  07ily  upon  the  2)Osition  of  the  mov- 
ing line,  and  is  the  same  whethej'  we  take  one  direction  or  the 
other  as  positive. 

282.  Division  of  the  Angle,  As,  in  §  280,  we  have  sup- 
posed the  point  x  to  divide  the  line  ah,  so  we  may  in  the  jire- 
ccding  construction  suppose  the  line  OX  to  divide  the  angle 
BOA  into  the  parts  BOX ^i\^  AOX.  We  then  take  for  the 
dividing  ratio,  not  the  ratio  of  the  angles  themselves,  but  that 
of  their  sines. 

Note.  The  student  may  remark  a  certain  incongruity  when  we 
speak  of  the  point  x  dividing  the  Hne  ab  into  the  segments  ax  and  bx, 
because  it  is  not  the  algebraic  sum  but  the  algebraic  difference  of  the 
8eii;ments  which  makes  up  the  Hne  ab.  This  incongruity  would  be 
avoided  by  measuring  one  of  the  segments  in  the  opposite  direction, 
making  x  its  initial  point,  thus  taking  ax  and  xb  as  the  segments.  But 
it  is  more  convenient  to  take  x  as  the  terminal  point  of  each  segment, 
and  to  accept  the  incongruity  of  calling  a  line  the  algebraic  difference 
of  its  parts,  because  no  confusion  will  arise  when  the  case  is  once  under- 
stood. 

The  same  remarks  apply  to  the  division  of  the  angle, 

283.  Distinction  of  Aiitecedent  and  Consequent  in  Dis- 
tance- and  Sine-Ratio.  In  forming  a  ratio  one  of  the  terms 
must  be  taken  as  the  antecedent  (or  dividend),  and  the  other 
as  the  consequent  (or  divisor).  By  interchanging  the  points  a 
and  b  the  antecedents  and  consequents  will  be  interchanged, 
and  the  ratio  will  therefore  be  changed  to  its  reciprocal. 

To  give  clearness  to  the  subject  we  shall  employ  the  fol- 
lowing notation: 

The  points  a  and  h  from  which  we  measure  the  segments 
ax  and  Ix  will  be  called  hase-points. 

That  base-point  from  which  the  antecedent  segment  of  the 
ratio  is  measured  will  be  called  the  A-point. 

That  base-point  from  which  the  consequent  segment  of  the 

ratio  is  measured  will  be  called  the  B -point. 

ax 
Then,  when  the  ratio  ~  is  represented  in  the  form 

{a,  I,  X), 


314  MODERN  GEOMETRY. 

we  write  first  the  A-point,  next  the  B-point,  and  lastly  the 
terminal  point,  which  we  may  call  the  T-point. 

284.  Permutation  of  Points.    If  we  use  the  notation 

jp  =  length  ab, 
q  El  length  hx, 


we  shall  have,  by  the  definition  of  the  distance-ratio. 

Let  us  represent  this  ratio  by  the  symbol  r.  Then,  by 
permuting  the  base-points  between  themselves  (that  is,  by 
making  a  the  B-  and  h  the  A-point),  we  shall  have 

{h,  a,  x)  =  — ^  =  i,  (J) 

a  result  which  we  may  express  by  the  general  proposition: 

I.  By  permuting  the  base-points  we  change  the  distance- 
ratio  into  its  reciprocal. 

By  permuting  b  and  x  in  («),  we  have 

(a,x,b)  =  f^  =  -^  =  l-r.  (c) 

That  is: 

II.  By  permuting  the  B-  and  T-points  we  change  the  dis- 
tance-ratio r  into  1  —  r. 

The  same  permutation  applied  to  {b)  gives 

(b,  x,  a)  =  —  =  — T— n^ — r  =  — n —  = .        (d) 

^  '    '    '       xa       —  {p  -i-  q)      P  -\-  q  r 

Lastly,  by  permuting  the  base-points  in  (c)  and  (c?), 

{x,  a,  b)  =  ^-—^;  (e) 

(a:,  b,  a)  =  — ^.  (/) 


DISTANCE-  AND  SINE-RATIO. 


315 


EXERCISES. 

1.  By  comparing  the  forms  {a)  and  (d),  show  that  if,  in 
the  expression 


cpr 


r  -  1 


we  put  q)r  for  r  and  repeat  the  substitution  in  the  result,  we 
shall  get  r  itself. 

2.  Find  the  distance  from  the  A-point  (§  277)  of  points 
whose  distance-ratios  are 


2'^2'^2* 


A        2 
Ans.  -p) 


-V\    ^P- 


3         1 

If  «,  h,  X  and  y  be  any  points  whatever,  show  that 
(a,  hy  x)  _  (i,  a,  y)  _  (x,  y,  a)  _  {y,  x,  h) 


4. 
of 


(a,  h,  y)  {h,  a,  x)  {x,  y,  h) 
Show  that  if  from  the  vertex 
an  isosceles  triangle  abc  we 
draw  a  line  ex  to  the  base,  the  sine- 
ratio  in  which  the  angle  c  is  divided 
by  the  line  ex  equals  the  distance- 
ratio  in  which  the  base  ab  is  cut  by 
the  point  x. 

In  algebraic  language  the  theorem  is 


a)- 


a^ 


sm  acx 


ax 
hx' 


sin  bcx 

5.  If  the  angle  A  OB  is  120°,  in  what  directions  must 
those  lines  be  drawn  which  will  divide  the  angle  in  the  respec- 
tive sine-ratios  —  2  and  +  2? 

6.  If  the  point  x  divide  the  segment  ah  in  the  ratio  -j-  1  :  2, 
in  what  ratio  will  h  divide  the  segment  ax^         Ans.  2  :  3. 

7.  If  the  points  x  and  y  divide  the  segment  ai  in  the  re- 
spective ratios  +  2  and  —  2,  in  what  ratios  will  a  and  b  re- 
spectively divide  the  segment  xy?  Ans.  +  3  and  —  3. 

8.  If  the  sum  of  the  distance-ratios  of  two  points,  x  and  y, 
is  unity,  show  that  ax  x  ay  —  ah"*. 


316 


MODERN  GEOMETRY. 


Theorems  involving  the  Distance-  and  Sine- 

Katios. 

385.  Def,  If  each  of  the  sides  or  angles  of  a  polygon  is 
divided  by  a  point  or  line,  the  ratios  of  the  divisions  are  said 
to  be  taken  in  order  when  each  vertex  is  a  divisor-point  for 
one  of  its  sides  and  a  dividend-point  for  the  other  side. 

If  the  divisions  are  all  internal,  we  shall,  in  going  round 
the  polygon,  have  the  divisor-  and  dividend-segments  in  alter- 
nation. 

386.  Theorem  I.  If  any 
three  lines  le  drawn  from  the 
three  vertices  of  a  triangle  to  its 
opposite  sides,  the  contiiiued 
product  of  the  sine-ratios  in 
which  the  angles  are  divided  isc' 
equal  to  the  continued  product 
of  the  distance-ratios  i7i  which 
the  sides  are  divided,  the  ratios 
leing  all  talcen  iyi  order. 

Hypothesis.  A  triangle  ahc  of  which  the  sides  and  angles 
are  divided  by  the  lines  ax,  ly  and  cz. 

Conclusion.     If  we  put 

r^,  r^,  7*3,  the  distance-ratios  in  which  the  sides  are  divided 
by  the  points  x,  y  and  z  respectively; 

R^,  R^,  R^,  the  sine-ratios  in  which  the  angles  are  divided 
by  the  respective  lines  ax,  ly  and  cz,  we  have 

^1  ^\  ^   =   ^1  ^.  ^3- 

Proof  By  the  theorem  of  sines  in  trigonometry  we  have, 
in  the  triangles  hax  and  cax, 

ex  _  sin  cax  ^ 
ax  ~    sin  c    ' 
hx  _  sin  hax 
ax  ~    sin  h 


DISTANCE-  AND  SINE-RATIO. 


317 


Dividing  the  first  equation  by  the  second, 
ex       sin  cax     sin  b 


In  the  same  way  we  find 


hx~ 

sin  bax 

sin  c 

find 

^Z- 

sin  aby 

sin  c 

cy 

sill  cby 

sin«' 

bz 

sin  bcz 
sm  «6';Z 

sin  a 

az 

sin  b' 

Taking  the  continued  product  of  the  three  last  equations,  we 
have 

ex     ay     bz  _  sin  eax     sin  aby     sin  bcz 

bx 


ay 
~cy 


az 


sin  bax     sin  eby     sm  acz 

The  three  fractions  in  the  first  member  of  this  equation 
are  the  distance-ratios  in  which  the  sides  are  divided,  and 
those  in  the  second  member  are  the  sine-ratios  in  which  the 
angles  are  divided,  so  that  the  theorem  is  proved. 

28 1*.  Theorem  II.  The  continued  product  of  the  dis- 
tance-ratios in  which  any  transversal  cuts  the  sides  of  a  tri- 
angle is  equal  to  unity. 

Proof  Let  a  transversal 
cut  the  sides  of  the  triangle 
abc  in  the  points  x,  y  and  z. 

Through   any  vertex,  as^ 
b,  draw  a  line  parallel  to  the 
transversal,  meeting  the  op- 
posite side  in  the  point  b'. 

Then,  forming  the  distance-ratios  in  which  the  sides  bo 
and  ba  are  divided,  using  the  similar  triangles  thus  con- 
structed, we  have 

az  _  ay 
Tz~t~y' 
bx  _  b'y  ^ 


ex 


while 


cy 


cy  ^  c/y_ 
ay      ay 


318  MODERN  GEOMETRY. 

The  continued  product  of  these  equations  gives 

££.*?..  oy.  ^  1.    Q.  E.  D. 
Dz     ex     ay 

Remaek.  Since  the  demonstration  takes  no  account  of 
algebraic  signs,  we  have  not  yet  shown  whether  the  product 
is  +  1  or  —  1.  It  is  evident  that  the  transversal  must  cut 
either  two  sides  of  the  triangle  internally,  or  none.  Hence 
either  two  factors  or  none  at  all  will  be  negative;  whence  the 
product  is  always  positive  and  equal  to  +  1. 

Corollary.  If  three  points  in  a  straight  line  he  tahen  on 
the  three  sides  of  a  triangle,  the  junction-lines  from  each  point 
to  the  opposite  ve7iex  divide  the  a7igles  into  parts  the  continued 
product  of  luhose  sine-ratios  is  imity. 

For,  by  Th.  I.,  the  product  of  the  sine-ratios  is  equal  to 
that  of  the  distance-ratios,  and,  by  Th.  11. ,  the  continued 
product  of  the  latter  is  unity. 

288,  Theorem  III.  Conversely,  If  on  the  three  sides 
of  a  triangle  abc  lue  take  any  three  j^oints  x,  ?/,  z,  such  that 

az     hx    cy  _ 

bz  '  ex  '  ay        ' 

these  points  will  he  in  a  straight  line. 

Proof  Let  z'  be  the  point  in  which  the  line  xy  cuts  the 
side  ah  of  the  triangle.     We  shall  then  have,  by  Th.  I., 

az'     hx     cy  _ 
hz'  '  ex'  ay  ~ 

Comparing  with  the  equation  of  the  above  hypothesis,  we 
find 

az  _  az\ 

Tz~  U'' 

that  is,  the  distance-ratios  of  the  points  z  and  z'  with  respect 
to  a  and  h  are  the  same. 

Because  there  is  only  one  point  on  ab  which  has  a  given 
distance-ratio,  the  points  z  and  z'  are  coincident  and  z  hes 
on  the  line  xy.     Q.  E.  D. 


DISTANCE-  AND  SINE-RATIO. 


319 


289,  Theorem  IV.     If  three  lines  2^assi7ig  through  a 
point  be  draiun  from  the  vertices  of  a  tri 
angle,  the  angles  will  be  so  divided  that  the 
sine- ratios,  taken  in  order,  will  be  —  1. 

Proof.  If  ABC  be  the  triangle,  and 
P  the  point,  we  have,  in  the  triangles 
PAB,  PBO  and  PCA,  neglecting  alge- 
braic signs, 

sin  BAP  _AP^ 
sin  ABP  ~  BP' 
sin  CBP  _BP^ 
sin  BCP~  OP' 
sin  ACP  _  CP 
sin  GAP  ~  AP' 


The  continued  product  of  these  equations  would  give 
sin^^P      sin  CBP     sin  ACP 


sin  CAP  '  sin  ABP  '  sin  BCP~ 


±1. 


(a) 


Algebraic  signs  having  been  neglected,  it  remains  to  be 
found  whether  this  product  is  positive  or  negative.  We  have 
the  theorems: 

I.  Lines  drawn  from  any  point  within  a  triangle  to  the 
three  vertices  cut  the  angles  internally. 

II.  Of  the  three  lines  drawn  to  the  vertices  of  a  triangle 
from  an  external  point,  and  produced  if  necessary,  two  will 
divide  the  angles  internally  and  one  externally. 

I  is  evident.  To 
prove  II  let  the  whole 
plane  without  the  tri- 
angle be  divided  by  its 
sides  into  the  six  regions 
A,A',B,B',  CandC". 
Then  the  angle  whose 
sides  bound  A  will  be 
cut  internally  or  exter- 
nally, according  as  the 
point  is  situated  within  or  without  one  of  the  regions  A  and 


320  MODERN  GEOMETRY. 

A\  Considering  the  otlier  angles  in  the  same  way,  we  see 
that  onl}^  one  angle  can  be  cut  internally  and  that  the  other 
two  will  be  cut  externally. 

The  sine-ratio  being  positive  for  an  external  and  negative 
for  an  internal  division,  either  one  or  all  three  of  the  factors 
in  {a)  must  be  negative.     Hence 

sin  BAP     sin  CBP     sui  A CP  _ 
sin  CAP  '  sin  ABP  '  sin  BCP  ~         *  Q.  E.  D. 
Corollary.     Three  lines  passing  from  the  vertices  of  a  tri- 
angle through  a  2^oint  cut  the  opposite  sides  so  tliat  the  con- 
tinued product  of  the  distance-ratios,  taken  in  order,  is  nega- 
tive unity. 

For,  by  Theorem  I.,  this  product  is  equal  to  that  of  the 
corresponding  sine-ratios,  which  product  is  negative  unity,  by 
the  theorem. 

390.  Theorem  V.  Conversely,  If  three  points  cut  the 
respective  sides  of  a  triangle  so  that  the  continued  product  of 
the  distance-ratios  is  negative  unity,  the  lines  joining  these 
2Joi7its  to  the  opposite  vertices  of  the  triangle  yass  through  a 
point. 

Proof  If  aic  be  the  triangle,  and  x,  y  and  z  be  the  points, 
we  have,  by  hypothesis, 

az     hx     cy  _ 
bz  '  ex  '  ay 
Join  the  points  x  and  y  to  the  opposite  vertices,  a  and  b, 
of  the  triangle  by  lines  intersecting  at  a  point  0.     From  c 
draw  a  line  L  through  0,  and  let  z'  be  the  point  in  which  it 
cuts  be. 

By  Th.  IV.,  Cor.,  we  then  have 
az'      bx       cy  _ 
bz'  '    ex  '   ay 
Comparing  with  the  hypothesis,  we  have 
az  _  az' 
Tz  ~  W 
Therefore  the  points  z  and  z'  are  coincident  and  the  line 
from  z  to  c  is  identical  with  L,  and  so  passes  through  the 
point  0  in  which  ax  and  by  intersect.     Q.  E.  D. 


Q 


ANHARMONIC  RATIO.  321 

EXERCISES. 

1.  Explain  wliiit  Tlicorem  11.  shows  when  the  transversal 
is  parallel  to  one  of  the  sides. 

2.  What  does  I'heorcni  IV.  become  when  the  point 
through  which  the  lines  are  drawn  is  at  infinity? 

3.  Show  by  Theorem  V.  that  the  three  medial  lines  of  a 
triangle  pass  through  a  point. 

4.  Show  by  the  preceding  theorems  what  bisectors  of  the 
interior  and  exterior  angles  of  a  triangle  meet  in  a  point. 

5.  If  from  the  vertices  at  the  ends 
of  the  base  BC  ot  a  triangle  we  draw 
lines  intersecting  on  the  medial  line 
AQ  and  meeting  the  opposite  sides  in 
the  points  B^  and  C,  show  that  B'C 
is  parallel  to  BC. 

6.  In  this  case  what  relation  exists B' 
between  the  distance-ratios  in  which  the  sides  AB  and^C 
are  divided  by  the  points  B'  and  C? 

The  Anliarmonic  Ratio. 

391.  Taking  any  point  x  on  a  J       t  y 

the  line  ab,  we  have,  by  what  precedes,  a  distance-ratio  ax  :  Ix 
or  (rt,  h,  X)  of  the  point  x  with  respect  to  the  jioints  a  and  h. 
In  the  same  way,  taking  a  fourth  point  y,  w^e  have  a  distance- 
ratio  («,  J,  y).     Then: 

id    1)   xS 

Def.     The  quotient  ;  '  / — f  of  the  distance-ratios  of  the 
(«,  ^,  y) 
points  X  and  y  with  respect  to  the  points  a  and  h  is  called  the 
anhariuonic  ratio  of  the  four  points  a,  J,  x  and  y. 

Tliat  terminal  point  x  which  enters  into  the  numerator  of 
the  fraction  wdll  be  called  the  A-T-point;  the  other,  the  B-T- 
point. 

It  will  be  seen  that  the  anharmonic  ratio  is  a  pure  number 
■whose  value  depends  upon  the  mutual  distances  of  the  four 
points. 

292.  The  following  are  simple  corollaries  from  the  de- 
finition of  the  anharmonic  ratio: 


822  MODERN  GEOMETRY. 

I.  If  the  terminal  points  are  ioth  outsiae  the  segment  ah, 
or  both  within  it,  the  anharmojiic  ratio  is  positive. 

For  in  the  first  case  the  distance-ratios  are  both  positive, 
and  in  the  second  they  are  both  negative. 

II.  If  one  terminal  point  is  witJmi  and  the  other  without 
the  segment  ah,  the  anharmonic  ratio  is  negative. 

For  the  two  distance-ratios  then  have  opposite  signs. 

III.  If  the  two  terminal  points  coincide,  the  anharmonic 
ratio  is  unity. 

IV.  If  three  points,  namely,  the  hase-points  and  one  ter- 
minal point,  are  fixed,  while  the  other  terminal  point  may  move, 
then  for  every  value  which  we  may  assign  to  the  anharmonic 
ratio  there  will  ie  one  and  only  one  position  of  the  movable 
point. 

For  if  we  put  r  —  the  anharmonic  ratio,  and  suppose  the 
points  a,  b  and  x  to  be  fixed,  we  have,  by  definition. 


whence 


{a,  b,  x)  ' 
(a,  h,  y)  =  (a,  b,  x)  X  r. 


Now,  the  points  a,  b  and  x  being  fixed,  the  quantity 
{a,  b,  x)  is  a  constant,  so  that  for  every  different  value  we 
assign  to  r  we  shall  have  a  different  value  of  the  distance- 
ratio  {a,  b,  y),  and  hence  a  different  position  of  the  point  y 
(§279). 

It  will  be  seen  that  the  four  points  which  enter  into  an 
anharmonic  ratio  form  two  pairs,  one  pair  being  the  two  base- 
points,  the  other  pair  the  two  terminal  points.  The  two 
points  of  each  pair  are  said  to  be  conjugate  to  each  other. 

ia    b  x'\ 
Notation".     We  represent  the  anharmonic  ratio  )  '  / — { 

(«,  ^,  y) 

in  the  form 

{a,  b,  X,  y). 

Expressing  the  points  by  their  general  designations 
(§§  283,  291),  the  order  of  writing  them  is 

(A,  B,A-T,B-T). 


ANHARMONIC  RATIO. 


Writing  the  ratios  at  length,  we  have 

,      ,  .       ax  '.  hx       ax  .by  .  . 

ia,  by  X,  y)  = -~  = —-.  (a) 

^  '    '    '  ^'      ay:  by      ay  .  bx  ^  ' 

293.  Permutation  of  Points.  Let  us  now  consider  the 
problem,  What  changes  will  result  in  the  anharnionic  ratio 
by  interchanging  the  different  points? 

By  interchanging  the  two  base-points,  that  is,  by  making 
b  the  A-point  and  a  the  B-point,  we  change  each  distance- 
ratio  into  its  reciprocal  (§  284,  I),  and  hence  the  anharmonic 
ratio  into  its  reciprocal,  because  we  always  have 

l:q      p' 

whatever  be  p  and  q. 

The  same  result  follows  by  interchanging  the  terminal 
points,  because  we  then  change  the  terms  of  the  fraction 
(a^^x)     .^^^     {a,  b,  y) 
(a,  b,  y)  {a,  b,  x)' 

Hence,  if  we  make  both  changes,  the  anharmonic  ratio 
will  be  restored  to  its  original  value. 

If  we  simply  make  the  base-points  the  terminal  ones,  and 
vice  versa,  the  anharmonic  ratio  is  unaltered.  For,  by  the 
notation, 

xa  :  ya  _  xa  :  xb 


{x,  y,  a,  b) 


xb  :  yb      ya  :  yb^ 


which  is  identical  with  (a),  the  signs  of  each  of  the  four  seg- 
ments being  changed. 

It  follows  from  this  that  there  will  be  four  permutations 
which  will  leave  the  anharmonic  ratio  unchanged,  and  four 
others  which  only  change  it  to  its  reciprocal.  They  are  as 
follows: 

{a,  b,  X,  y)  =  {b,  a,  y,  x)  =  {x,  y,  a,  b)  =  {y,  x,  b,  a)  =  r;  (1) 
(5,  a,  X,  y)  =  {a,  b,  y,  x)  =  {x,  y,  b,  a)  =  (y,  x,  a,  b)  =  -.(2) 


324  MODERN  GEOMETRY. 

In  all  these  permutations  the  four  points  are  paired  in  the 
same  way,  a  and  b  being  one  pair  and  x  and  y  the  other. 

Hence  the  eight  permutations  which  do  not  change  the 
pairing  of  the  conjugate-points  can  only  interchange  the  terms 
of  the  anharmonic  ratio. * 

When  the  pairing  of  the  points  is  changed,  a  may  ha-ve 
either  x  or  y  as  its  conjugate-point.  To  find  the  effect  of 
these  permutations,  we  start  from  the  following  identical 
equation  which  always  subsists  between  the  six  segments  ter- 
minated by  the  four  points  «,  h,  x  and  y.  These  segments 
are  al),  ax,  ay,  bx,  by  and  xy.  h h + j^ 

ax .  by  +  <^^  •  y^  +  <^y  •  ^^  =  ^'  (^) 

To  proYG  this  equation,  Ave  substitute  for  ab  and  ay  their 
values 

ab  =  ax  +  ^bf 

ay  =  ax  -\-  xy, 

and  so  write  the  first  member  of  the  equation  in  the  form 

ax.by  -\-  ax    yx  -\-  ax    xl), 

-\-  xb        +  ^y 
which  is  the  same  as 

ax{by  -i-  yx  -\-  xb)  +  xb{yx  +  xy), 
an  expression  which  vanishes  identically,  because 

by  -\-  yx  +  ^^  =  0;     yx  -{-  xy  E  0. 

Now  divide  (a)  by  ay  .bx.     We  thus  find 

ax .  by       ab .  yx  _     ^ 
ay.bx      ay  .bx  ~    ' 
that  is, 

(a,  b,  X,  y)  -f  {a,  x,  b,  y)  =  I.  (b) 

Hence,  using  the  same  notation  as  before, 

(a,  X,  b,  y)  =1-  r. 

*  In  tills  pairing  process  note  the  analogy  of  conjugate-points  to 
partners  at  -whist.  There  are  eight  arrangements  of  the  players  around 
the  table  which  will  not  change  the  pairing,  and  there  are  three  ways 
in  which  the  players  may  choose  partners. 


ANHARMONIC  RATIO.  325 

We  now  liave,  in  the  same  way  as  before, 

(«,  X,  h,  y)  =  {x,  a,  y,  b)  =  (b,  y,  a,  x)  =  (y,  b,  x,  a)  =  1  -  r;  (3) 

{x,  a,  b,  y)  =  (a,  x,  y,  b)  =  {y,  b,  a,  x)  =  {b,y,  x,  a)  =  ^--— -.  (4) 

We  have  finally  to  consider  the  case  in  which  a  is  paired 
with  y.  To  pair  a  with  y,  we  remark  that  the  equation  [b), 
being  true  whatever  points  we  suppose  a,  b,  x  and  y  to  rep- 
resent, may  be  considered  a  brief  expression  of  the  theorem: 

By  inter clianging  the  B-  and  A-T-i)oints,  we  form  a  new 
anharmonic  ratio  tuliich,  added  to  the  original  one,  makes 
unity. 

Applying  this  theorem  to  the  second  expression  in  line  (4), 

it  gives 

(«.,  X,  y,  b)  +  {a,  y,  x,  b)  =  1. 

1  r 

Hence    {a,  y,x,b)  —  1  —  {a,  x,  y,  b)  =  1  — = 


r      r  —  1' 
and 

{a,  y,  X,  b)  =  {y,  a,  b,  x)  =  {b,  x,  y,  a)  =  {x,  b,  a,  y)  =  ^r^;  (5) 

r  —  1 
(y,  a,  X,  b)  =  (a,  y,  b,  x)  =  {x,  b,  y,  a)  =  {b,  x,  a,  y)  =  —^.  (6) 

The  equations  (1)  to  (6)  include  all  24  permutations  of 
a,  b,  X  and  y,  which,  however,  give  rise  to  only  G  different 
values  of  the  anharmonic  ratio,  namely, 

1  _  1  r  r  —  1  ,  . 

r  \  —  r  r  —  1  r 

294.  The  preceding  operations  lead  to  a  curious  algebraic 
result.  Suppose  that,  instead  of  starting  with  the  equation 
(1),  we  had  started  with  any  of  the  others,  (G)  for  example. 
We  could  then  have  obtained  expressions  for  the  remaining 
20  anharmonic  ratios  by  performing  the  same  operations  on 
(6)  which  we  have  actually  performed  upon  (1),  only  the  ex- 
pression  would  have  taken  the  place  of  r  all  the  way 

through.   But,  if  the  process  is  correct,  we  should  then  arrive  at 
the  same  expressions  for  the  other  20  anharmonic  ratios  which 


326  MODERN  GEOMETRY. 

weliave  actually  found.  The  same  being  true  if  we  start  from 
any  other  of  the  six  equations,  we  conclude: 

If,  in  the  set  of  expressions  (c),  we  substitute  for  r  any  one 
expression  of  the  set,  the  values  of  the  several  expressions  zvill 
he  cha7iged  into  each  other  in  such  a  way  that  the  set  will  re- 
main unaltered  except  in  its  arrangement. 

As  an  illustration,  let  us  substitute  the  sixth  expression, 

r  —  \ 

,  for  r  all  the  way  through  the  set  (c).  Then,  by  reduc- 
tion of  the  fractions, 

r  —  1 

r  will  be  changed  to 


r 


1  r 

—  will  be  changed  to 


r  r  —  1' 

1  —  r  will  be  changed  to  — ; 

will  be  changed  to  r; 

-  will  be  changed  to  1  —  r; 

will  be  changed  to . 


r 
r  — 
r-1 


We  have  thus  reproduced  the  same  set,  only  differently 
arranged. 

295.  Anharmonic  Ratio  of  a  Pencil  of  Lines.  As  we 
have  formed  the  anharmonic  ratio  of  four 
points  on  a  line  by  their  distance-ratios,  so 
we  may  form  the  anharmonic  ratio  of 
four  lines  passing  through  a  point  by 
means  of  their  sine-ratios. 

The  four  lines  A,  B,  X  and  Y  pass    .  i 
through  the  point  P.     If  we  take  A  and  \^ 

B  as  the  base-lines  forming  the  angle  APB,  the  line  X  will 
give  the  sine-ratio 

BmAPX 
liiiBPX' 


Aiikar^(pulc  ^aY/os^ 


V^ 


P^'P^' P^<^  Si^x  0(   ^JUy<5' 


AP'  ^  8V^-    ^^c^(,^^     ^ 
OA        03  OP  oP  ^ 


z. 


K 


JX      Xt^O<> 

Ax    _  AI^  .  BP 


7<^ 


ANHARMONIC  RATIO. 


327 


and  the  line  Fwill  give  the  sine-ratio 

sin  APY 
sin  BPY' 

The  quotient  of  these  ratios,  or 

sin  APX  .  sin  APY  _  sin  .^PX  sin  BPY 
sin  BPX  '  sin  BPY  ~  sin  BPX  sin  APY' 

is  called  the  anharmouic  ratio  of  the  pencil  of  lines  PA, 
PB,PXaiidPY. 

Designating  each  line  by  a  single  letter,  we  may  write 

(A,  B,  X,  Y) 

as  the  anharmonic  ratio  of  the  four  lines  A,  B,  Xand  Y. 

296.  FuKDAMENTAL  THEOREM.  If  a  transversol  cvoss 
a  pencil  of  four  lines,  the  anharmonic  ratio  of  the  four  points 
of  intersection  icill  he  equal  to  the  anhar^nonic  ratio  of  the 
pencil. 

Proof  Let  ABXYhQ  the 
pencil,  intersecting  the  trans- 
versal ay  in  the  points  a,  l,  x 
and  y. 

We  begin,   as  in  §  286,  by 
comparing   the   distance-ratio" 
{a,  h,x)  with  the  sine-ratio  {A,  B,  X). 

From  the  equations 

ax  :  Px  =  sin  aPx  :  sin  xaP^ 
bx  :  Px  =  sin  hPx  :  sin  xbP, 

we  obtain,  by  division, 

ax  _  sin  aPx     sin  xbP^ 
bx  ~  sin  bPx  '  sin  xaP' 

or,  using  the  abbreviated  notation, 

sin  xbP 


{a,  b,  x)  =  (A,  B,  X) 


sin  xaP' 


328  MODERN  GEOMETBT. 

We  find  in  the  same  way 

Taking  the  quotient  of  these  equations, 

(a,  h  a)  _  (^,  B,  X) 
K  b,  y)       (A,  B,  YY 

The  first  member  of  this  equation  is,  by  definition,  the 
anharmonic  ratio  of  the  four  points  a,  h,  x  and  ?/,  while  the 
second  is  that  of  the  pencil  of  lines  A,  B,  JTand  Y.     Thence 

(a,  h,  X,  y)  =  {A,  B,  X,  Y). 

Q.  E.  D. 

Cor.  1.  If  any  number  of  transversals  cross  the  same 
pencil  of  four  lines,  the  anharmonic  ratios  of  the  four  ^^oints 
of  intersection  on  the  several  transversals  will  all  be  equal. 

For  each  such  ratio  will  be  equal  to  the  anharmonic  ratio 
of  the  pencil. 

Cor.  2.  If  from  a  roiv  of  four  points  lines  be  drawn  to  a 
fifth  movable  point,  the  anharmonic  ratio  of  the  pencil  thus 
formed  ivill  be  C07istant,  lohatever  be  the  position  of  the  fifth 
point. 

For  the  anharmonic  ratio  of  the  pencil  will  be  constantly 
equal  to  that  of  the  row. 

Scholium.  Using  the  notation  of  §  276,  IX.,  the  preced- 
ing propositions  may  be  expressed  as  follows: 

If  P  be  any  vertex,  and  a,  b,  x,  y  a  row  of  points,  then 

Anh.  ratio  (P-a,b,x,y)  =  (a,  b,  x,  y). 

If  p  be  any  transversal,  and  A,  B,  X,  Y  the  four  lines  of 
a  pei^cil,  then 

Anh.  ratio  {p-A,B,X,Y)  =  {A,  B,  X,  Y). 

297.  Application  of  the  Principle  of  Duality.  The 
branch  of  Plane  Geometry  which  we  are  now  treating  is 
subject  to  the  principle  of  duality  (§  276);  that  is, 

From  every  proposition  respecting  the  relations  of  points 
and  lines  we  may  form  a  second  correlative  proposition  respect- 


ANUARMONIG  PROPERTIES.  329 

ing  the  relation  of  lines  and  points,  by  interchanging  the  words 
line  and  point,  as  follows: 

Line  instead  of  Point. 

Junction-point  of )              ^^  (Junction-line   of   two 

two  lines  )  \      points. 

Point  on  a  line  '^  Line  through  a  point. 

Pencil  of  lines  "  Row  of  points. 

Three  points  in  a )  ^  j  Three  lines  through  a 

line  )  (      point. 

Anharmonic  ration  (  Anharmonic   ratio  of 

of  hues  throudi  V  ''  ■{  -   ,  ,. 

.    ,  °    I  ]      points  on  a  line. 

a  point  ;  ( 

The  correlative  proposition  is  not  necessarily  different 
from  the  original  one.  When  the  two  are  identical,  the  pro- 
position is  self-correlative. 

The  relation  of  the  proposition  to  its  correlative  is  mutual; 
that  is,  the  correlative  of  the  correlative  is  the  original  pro- 
position. 

To  make  the  notation  correlative  we  represent  the  junction- 
point  of  the  two  lines  A  and  B  by  AB. 

Let  us,  as  an  example,  change  the  preceding  fundamental 
proposition  into  its  correlative.     The  two  then  read: 

^.  ( pencil  of  four  lines       t  r.<.,i    (line. 

Given ;  a  -^  ^         „  ^  .        and  any  fifth  ■<      ,  ' 

(  row  01  four  points  (  point, 

TVe  conclude :  the  anharmonic  ratio  of  the  four 

jnnction-  \  ^"'^  of  the  four  \  ^'^.^^    with  tlie  fifth  \  ''".<^ 
( lines  (  points  (  point 

equal  to  that  of  the  given  \  P®^^^  • 

(  row. 

By  reading  the  top  lines  Ave  have  the  original  proposition; 
l)y  reading  the  bottom  lines,  its  correlative.  By  making  the 
construction  it  will  be  seen  that  the  correlative  proposition  is 
identical  with  the  original  one. 

The  principle  of  duality  applies  to  the  demonstration  as 
well  as  to  the  proposition.  By  making  the  above  substitu- 
tions the  demonstration  of  the  original  becomes  the  demon- 


is 


330 


MODERN  GEOMETRY. 


stration  of  the  correlative.  It  is  tlierefore  in  rigor  not 
necessary  to  give  the  latter;  and  when  we  do  so,  it  is  only  to 
assist  the  student. 

398.  Theorem.  If  tve  have  two  lines  intersecting  m  a 
point  p,  and  if  we  have  on  the  one  line  any  three  points 
a,  h,  c,  and  on  the  other  line  three  points  a',  b',  c' ,  such  that 
the  anharmonic  ratio  (p,  a,  b,  c)  is  equal  to  (p,  a',  b',  c'), 
then  the  three  junction-lines   aa',  bb'  and  cc'  meet  in  a  point. 

Proof.  Let  if  and  iVbe  the  given  lines,  and  let  q  be  the 
junction-point  of  the  lines  aa'  and  bb\  Join  qp  and  qc',  and 
let  c"  be  the  junction-point  of  the  line  iV^  with  the  line  qc\ 


Then,  because  M  and  N  are  two  transversals  crossing  the 
pencil  q-a',p,h',c',  we  have 

[p,  a,  b,  c")  =  {p,  a\  b\  c'). 

By  hypothesis, 

(p,  a,  b,  c)     =  (p,  «',  b%  c'). 
Hence 

(p,  a,  b,  c")  =  {p,  a,  b,  c). 

The  points  j9,  a  and  b  being  given,  there  is  only  one  fourth 
point  which  can  form  with  them  a  given  anharmonic  ratio 
(§  292,  IV.).  Hence  the  points  c  and  c"  are  coincident, 
and  the  junction-line  c'c  is  identical  with  c'c",  and  so  passes 
through  the  point  q,  in  which,  by  construction,  aa'  and  bb' 
intersect.  Q.  E.  D. 


ANHARMONIC  PROrEBTIES.  331 

Correlative  Theorem.  If  we  have  two  pomfs  on  a  line 
Q,  and  if  tlivougli  one  point  pass  three  lines  A,  B  and 
C,  and  through  the  other  point  pass  three  lines  A\  B*  and  C", 
such  that  the  anharmonic  ratio  (Q,  A,  B,  C)  equals 
{Q,  A',  B'f  C),  then  the  three  junction-points  A  A',  BB'  and 
CC  lie  in  a  straight  line. 

Proof  Let  m  and  n  be  the  points;  let  P  be  the  junction- 
line  of  the  points  A  A'  and  BB'y  c  the  junction-point  PC\ 
and  C"  the  junction-line  nc. 


Then,  because  the  pencils  Q,  A*,  B\  C  and  Q,  A,  B,  C" 
pass  through  the  same  four  points  of  the  line  P,  we  have 

{Q,A,B,  0")  =  (Q,A',B',C"). 

But,  by  hypothesis, 

(Q,A',£',C")  =  (Q,A,B,C). 
TTen  oe 

(Q,A,B,Cn  =  {Q,A,B,  C) 

These  equal  anharmonic  ratios  having  three  lines  identical, 
the  fourth  lines  C  and  (7"  are  also  identical;  whence  the  lines 
C  and  C"  intersect  at  c,  the  junction-point  C^P;  whence  the 
junction-points  AA\   BB'  and   CC"  all  lie  on  the  line  P, 

Q.  E.  D. 

Note.  The  student  should  compare  the  demonstration,  step  by 
step,  with  that  of  the  original  proposition,  and  note  the  relation  of  each 
step. 


332  MODERN  GEOMETRY. 

Projective  Properties  of  Figures. 

299.  Let  there  be  a  j^oint  0,  a  plane  P  and  a  figiire  Q 
each  situated  in  any  position  in  space.  If  lines  (called  lines 
of  projection)  pass  from  0  to  each  point  of  Q,  the  points  in 
which  these  lines  intersect  the  plane  P  form  a  second  figure 
which  is  called  the  projection  of  the  figure  Q. 

This  definition  of  a  projection  is  more  general  than  that  of  elemen- 
tary geometry,  in  which  the  lines  of  projection  are  all  parallel  to  each 
other  and  perpendicular  to  the  plane  P.  The  latter  is  a  special  case  in 
which  the  point  0  is  at  infinity  in  a  direction  perpendicular  to  the  plane. 

It  may  be  remarked  that  the  shadow  of  a  figure  upon  a  plane,  as 
cast  by  a  luminous  point,  is  identical  with  its  projection.  But  should 
the  distance  of  any  part  of  the  figure  from  the  plane  exceed  the  distance 
of  the  luminous  point,  there  could  be  no  shadow,  but  there  would  still 
be  a  projection,  formed  by  continuing  the  lines  of  the  rays  in  the 
reverse  direction,  namely,  from  the  figure  through  the  luminous  point. 

300.  The  following  are  some  simple  relations  between 
figures  and  their  projections: 

I.  The  projection  of  a  point  is  a  point. 

II.  The  projection  of  a  straight  line  is  a  straight  line. 
For  since  the  straight  line  and  point  lie  in  a  plane,  the 

lines  of  projection  are  all  in  this  plane,  and  the  projection  is 
the  intersection  of  this  plane  with  the  plane  of  projection. 

III.  The  projection  of  a  roiu  ofpoi^its  is  another  row  ivhose 
carrier  is  the  projection  of  the  original  carrier, 

IV.  The  projection  of  a  pencil  is  a  2^e7icil. 

V.  The  projection  of  a  curve  and  a  tangent  is  another 
curve  and  a  tangent. 

VI.  Every  2^rojection  of  a  line  passes  through  the  point  in 
which  the  line  intersects  the  plane  of  projection. 

VII.  The  projection  of  a  circle  is  a  conic  section. 

For  the  lines  from  a  point  to  the  circumference  of  a  circle 
form  the  elements  of  a  cone.  Hence  their  intersection  with 
the  plane  of  projection  is  the  intersection  of  a  conical  surface 
with  that  plane,  and  is  therefore,  by  definition,  a  conic  section. 

VIII.  If  the  projected  figure  Q  is  in  a  plane  P',  and  if  we 
call  Q'  its  projection  on  the  plane  P,  then  Q  itself  is  the  pro- 
jection of  Q'  upon  the  plane  P'. 


ANHARMONIG  PROPERTIES. 


333 


This  follows  at  once  from  the  definition,  the  lines  of  pro- 
jection being  identical  in  the  two  cases. 

IX.  Every  section  of  a  circular  cone  can  he  projected  into 
a  circle. 

For,  by  taking  the  vertex  of  the  cone  as  tlie  point  0,  and 
its  circular  base  as  the  plane  of  projection,  the  outline  of  this 
base  becomes  the  projection  of  any  section  of  the  cone. 

301.  Theorem.  The  projection  of  a  roiu  of  foicr  ptoints 
has  the  same  anhar7nonic  ratio  as  the  original  row. 

Proof,  The  lines  of  projection 
of  the  four  points  form,  by  defini- 
tion, a  pencil  having  its  vertex  at 
0.  The  carriers,  both  of  the  origi- 
nal and  the  projected  row,   form 

transversals   crossing   this    pencil,  a' V c'  a' 

and  the  two  rows  of  points  are  the  intersections  of  these 
carriers  with  the  lines  of  the  pencil.  The  anharmonic 
ratios  of  the  two  rows  are  therefore  equal  (§  296,  Cor.  1). 

Q.  E.  D. 

302.  Theorem.  The  projection  of  a  pencil  of  four  lines 
has  the  same  anharmo7iic  ratio  as  the  original  pencil. 

Proof     Let    0-ahcd    be    the  ^0 

given  pencil,  and  let  a,  h,  c  and  d 
be  the  points  in  which  it  intersects 
the  plane  of  projection. 

Because  the  lines  of  the  pencil 
must  all  lie  in  one  plane,  the 
points  a,  h,  c  and  d  will  lie  in  a 
straight  line. 

If  0'  be  the  projection  of  0,  the  projected  pencil  will  be 
O'-ahcd. 

Then  the  anharmonic  ratios  of  each  pencil  will  be  equal 
to  («,  1),  c,  d),    and  so  will  be  equal  to  each  other. 

Remark.  Those  properties  and  relations  of  a  figure 
which  remain  unchanged  by  projection  are  called  projective 
properties  and  relations. 


334  MODERN  GEOMETRY. 


Harmonic  Points  and  Pencils. 

303.  Def.  When  the  anharmonic  ratio  of  four  points  is 
negative  unity,  tliey  are  called  a  row  of  four  harmonic 
points,  and  each  pair  of  conjugate  points  is  said  to  divide 
the  segment  joining  the  other  pair  harmonically. 

So  a  pencil  of  four  lines  of  which  the  anharmonic  ratio  is 
negative  unity  is  called  an  harmonic  pencil. 

Cor.  The  anharmonic  ratio  being  negative,  one  of  the 
terminal  points  must  divide  the  base-line  internally  and  the 
other  must  divide  it  externally.  Hence  the  order  of  the  four 
points  is  such  that  the  conjugate  points  of  the  one  pair,  a  and 
l,  alternate  with  those  of  the  other  pair,  x  and  y. 

If  the  point  x  is  half  way  between  a  and  h,  its  conjugate, 
y,  is  at  infinity. 

If  X  then  move  toward  h,  y  will  also  move  toward  h  from 
the  right,  and  the  two  points  will  reach  h  together. 

If  X  move  toward  a,  y  will  approach  from  infinity  on  the 
left,  and  the  two  points  will  reach  a  together. 

The  law  of  change  is  expressed  by  the  following  theorem: 

304.  Theokem.  The  product  of  the  distances  of  two  ter- 
minal harmonic  points  from  the  middle  of  the  base-line  is  con- 
stant, and  eqtial  to  the  square  of  half  the  base-line. 

Proof.  The  condition  that  the  anharmonic  ratio  of  the 
four  points  a,  b,  x,  y  =  —  1\b 

ax  ^   ay  _ 

bx  '   by  ~        ' 

which  is  equivalent  to 

ax.by  -\-  ay . bx  =  0. 

Let  m  be  the  middle  point  of  ab.     Then 

ax  =  am  +  mx; 

by  =■  bm-\-  my  =  —  am  -\-  my, 

ay  =  am -\- my; 

bx  =  —  am  -\-  nix. 


\ 


HARMONIC  POINTS  AND  LINES.  335 

By  substitution  tlie  equation  (a)  reduces  to 

—  2{a?ny  +  ^mx.my  =  0. 
Hence 

mx.my=(am)\  (1)     Q.  E.  D. 

Eemark.     On  the  line  ab  we  may  take  any  number  of 


' ' — I ,   I  „  i  .„' — nr, rr, 1' 

X    X    z  y  y  y 

pairs  of  points,  x  and  y,  fulfilling  the  condition   (1),  and 
therefore  dividing  harmonically  the  segment  ab. 

Def.  Three  pairs  of  points  which  divide  harmonically  the 
same  segment  are  said  to  form  an  involution. 

305,  The  Fourth  Harmonic. 

Def.  When  three  points  of  an  harmonic  row  are  given,  the 
fourth  is  called  the  fourth  harmonic  of  the  other  three. 

Problem.  Having  given  three  poi7its  of  an  harmonic  roiv, 
to  find  the  fourth. 

Construction.  '  Let  a,  h  and  c  be 
the  given  points,  and  let  a  and  h  be 
the  conjugate  base-points. 

On  the  middle  point  rn  of  ai 
erect  a  perpendicular  mp  =  \ah,  and  '^ 
on  the  other  side  of  m  from  c  take  mc'  =  mc. 

Through  j9  and  c'  describe  a  circle  having  its  centre  upon 
the  line  ab. 

The  other  point,  d,  in  which  this  circle  cuts  ab  will  be  the 
fourth  harmonic  required. 

Proof.  From  eq.  (1)  and  from  the  theorem  of  elementary 
geometry  which  gives  c'm.md  =^  {mpY  the  proof  is  readily 
found. 

306.  Fourth  Harmonic  of  a  Pencil.  When  three  lines 
of  a  pencil  are  given,  the  fourth  line  necessary  to  form  an 
harmonic  pencil  may  be  constructed. 

From  §  296  it  follows  that  every  pencil  of  four  lines  passing 
through  a  row  of  four  harmonic  points  is  an  harmonic  pencil, 
and,  conversely,  that  every  transversal  intersects  an  harmonic 
pencil  in  four  harmonic  points. 


336  MODERN  GEOMETRY. 

Hence,  to  construct  the  fourth  harmonic  of  a  given  pencil 
of  three  lines,  we  draw  any  transversal,  find  the  fourth 
harmonic  of  the  three  points  of  intersection,  and  join  it  to 
the  vertex. 

30')'.  Harmonic  Properties  of  the  Triangle, 
Theorem.     If  from  a  point  are  drawn  three  lines  to  the 
vertices  of  a  triaiigle,  and  at  any  two  of  the  vertices  the  fourth 
harmonics  to  the  lines  thence  emanating  are  constructed,  these 
fourth  harmonics  luill  rneet  C/ 

the  line  from  the  third  ver- 
tex in  a  point.  I       \    \^  '"*""'';^p' 

Proof  Let  ABC  be 
the  triangle,  and  P  the 
point.     Let  CP'  and  BP' 

be  the  fourth  harmonics  of  A^^^^ ^^^=^'B 

the  pencils  C-APB  and  B-APG. 

Then,  because  AP,  BP  and  CP  meet  in  a  point, 

sin  BAP     sin  CBP     sin  ACP  _  _  ..        . 

sm  CAP  '  sin  ABP  '  sin  BCP  '  ^^      '  ^ 

Because   BP^  and   CP'  are  fourth  harmonics  conjugate  to 
BP  and  CP  respectively, 

sin  CBP  _  _  sin  CBP\ 
sin  ABP  ~  sin  ABP'' 
sm  ACP  smACP' 


sin  BUP  sin  BCP'' 

By  substitution  in  the  equation  {a)  we  have 
sm  BAP     sin  CBP'     sin  ACP' 


=  -  1. 


sin  CAP  '  sin  ABP'  '  sin  BCP' 

Therefore  the  three  lines  AP,  BP'  and  CP'  meet  in  a 
point  (§§286,  290).     Q.  E.  D. 

Scholium.  By  drawing  the  fourth  harmonic  at  A  and 
considering  the  other  two  pairs  of  vertices,  A,  B  and  C,  A,  we 
have  two  other  points  of  meeting,  making  four  in  all. 

If  the  point  P  is  the  centre  of  the  inscribed  circle,  the 
lines  from  P  will  be  the  bisectors  of  the  angles,  and  the 
three  points  P'  will  be  the  centres  of  the  escribed  circles. 


HARMONIC  POINTS  AND  LINES. 


337 


308.  Correlative  Theorem.  If  three  points  on  a  line 
he  taken  07i  the  sides  of  a  triangle,  and  the  fourth  harmonics 
to  two  of  them  be  constructed,  these  fourth  harmonics  loill  he 
on  a  line  luith  the  third  point. 

Proof.     Let  ahc  be  the  triangle; 

X,  y  and  z,  three  points  on  the  sides  in  a  right  line; 

x',  y'  and  z' ,  the  fourth  harmonics  to  the  rows  b,  c,  x; 
c,  a,  y;  a,  h,  z,  respectively. 


Because  x',  y'  and  z'  are  fourth  harmonics, 

7   "^ 


az' 

hx^^ 

cy' 


iP) 


az  _ 

hz~ 

hx  _ 

ex  ~ 

cy  _ 

ay  ~        ay 
Because  x,  y  and  z  are  on  a  right  line, 
az     hx      cy  _ 
bz  '  ex  '   ay~ 

By  substituting  for  any  two  of  these  factors  their  values 
from  {b),  we  prove  the  theorem,  by  §  288. 

EXERCISES. 

1.  Any  two  orthogonal  circles  cut  the  line  joining  their 
centres  in  four  harmonic  points. 

2.  Any  circle  of  a  family  having  a  common  radial  axis 
cuts  harmonically  the  common  chord  of  the  orthogonal 
family. 


338  MODERN  OEOMETRT. 

Aiiliarmonic  Properties  of  Conies. 

309.  Lemma.  If  two  tangents  to  a  circle  are  intersected 
hij  a  third  tangent,  the  points  of  intersection  subtend  from 
the  centre  of  the  circle  an  angle  riieasured  by  one  half  the  arc 
letioeen  the  tioo  tangents.  p       \y 

Proof.     Let  0  be  the  centre 
of  the  circle; 

m,  n,  the  points  of  tangency 
of  the  two  tangents;  | 

p,  the   third  point   of    tan- 
gency; 

m',  ?i',  the  points  of  intersec-        .  . 

tion.  / 

It  is  then  easily  shown,  from  the  equality  of  the  lines  m'p 
and  m'm,  that  Om'  is  perpendicular  to  pni. 

In  the  same  way,  0?i^  is  perpendicular  to  p7i; 

Angle  m'  On'  =  angle  mp7i. 

By  a  fundamental  property  of  the  circle. 

Angle  mp}i  =  -J  angle  7uOn. 
Therefore 

Angle  m' On'  =  i  angle  ?nOn. 

Q.  E.  D. 
Cor.     If  the  third  tangent  pn'  moYes  around  the  circle, 
the  angle  m' On'  will  remain  constant,  being  always  equal  to 
\m  On. 

310.  Theorem.  If  fonr  fixed  tangents  touch  a  conic, 
and  a  inovable  fifth  tangent  intersect  them,  the  anliarmonic 
ratio  of  the  four  points  of  intersect  ioji  is  the  same  for  all  posi- 
tions of  the  fifth  tangent. 

Proof.     Let  the  conic  be  projected  into  the  circle  whose 
centre  is  0,  and  let  a,  h,  c  and  d  be  four  points  in  which 
the  fifth  tangent  intersects  four  fixed   projected  tangents, 
touching  the  circle  at  the  points  t^,  t^,  t^,  t^. 
Because  Angle  aOb  =  i  arc-angle  tf^,  j 

Angle  bOc  =  i  arc-angle  tj,   V  (g  309) 

and  Angle  cOd  —  ^  arc-angle  tj^,  ) 


ANUABMONIG  PROPERTIES  OF  CONICS. 


339 


and  t^,  /„  etc.,  are,  by  hypothesis,  fixed,  these  angles  remain 
constant  however  the  fifth  tangent  may  move. 


Therefore  the  anharmonio  ratio  of  the  pencil  0-abcd  re- 
mains constant,  being  a  function  of  the  sines  of  constant 
angles. 

Therefore  the  anharmonic  ratio  of  the  row  a,  h,  c,  d  re- 
mains constant  (§296). 

Because  this  anharmonic  ratio  is  constantly  equal  to  that 
of  the  corresponding  points  in  the  projected  figure  (§  301), 
the  latter  also  remains  constant.  Q.  E.  D. 

311,  Theorem.  If  from  foiir  fixed  points  of  a  come 
lines  le  draimi  to  a  fifth  variable  point,  the  anharmonic  ratio 
of  the  pencil  thus  formed  will  remain  constant  whatever  the 
position  of  the  fifth  point. 

Proof.  Project  the  conic  into  a  cir- 
cle. Let  a,  hy  c  and  d  be  the  projec- 
tions of  the  four  fixed  points,  and  P 
that  of  the  fifth  point. 

By  a  fundamental  property  of  the 
circle  the  angles  aPh,  hPc,  etc.,  will  re- 
main constant  as  P  moves  on  the  circle. 

Therefore  the  anharmonic  ratio  of 
the  pencil  P-abcd  will  remain  constant. 

Therefore  the  anharmonic  ratio  of  the  corresponding  pen- 
cil in  the  conic  will  remain  constant  (§  302).     Q.  E.  D. 

Dff.     The  constant  anharmonic  ratio  of  a  pencil  whose 


340  MODERN  OEOMETRT. 

lines  pass  from  four  fixed  points  on  a  conic  to  any  fifth  point 
is  called  the  anharmoiiic  ratio  of  the  four  points  of  the  conic. 
Def.  The  constant  anharmonic  ratio  of  the  points  in  which 
four  fixed  tangents  to  a  conic  intersect  a  fifth  tangent  is  called 
the  anliarmo7iic  ratio  of  the  four  tangents  to  the  co7iic. 

312.  Extension  of  the  Principle  of  Duality  to  Ctirves. 
If  we  conceive  a  series  of  points  to  follow  each  other  according 
to  some  law,  their  junction-points  will  form  a  broken  line  or 
a  polygon.  If  each  point  of  the  series  approaches  indefinitely 
near  to  the  preceding  one,  the  broken  line  approaches  a  curve 
as  its  limit.  We  may  therefore  define  a  curve  as  the  limit  of 
a  series  of  junction-lines  when  the  points  approach  each  other 
indefinitely. 

The  correlative  conception, 
on  the  principle  of  duality, 
will  be  that  of  a  series  of  lines 
following  each  other  according 
to  some  law,  and  approaching 
each  other  indefinitely.  The 
junction-points  of  consecutive^ 
lines  will  be  the  correlatives  of 
the  broken  lines,  and  as  they  approach  each  other  indefinitely 
they  will  tend  to  lie  on  some  curve  as  their  limit. 

In  the  first  case,  if  we  suppose  the  points  to  be  consecu- 
tive positions  of  a  moving  point,  this  point  will  move  on  the 
limiting  curve. 

In  the  correlative  case,  if  we  suppose  the  lines  to  be  the 
consecutive  positions  of  a  moving  line,  this  line  will  con- 
stantly be  tangent  to  the  limiting  curve. 

We  thus  have,  as  correlative  conceptions: 

Points  on  a  curve      corresponding  to      Tangents  to  a  curve. 

Junction-point  of  )  ^  ..     j  Junction-line  of  two 

two  tangents      )  (    points,  i.e.,  a  chord. 

Points  in  which  a)  (  rn  .        ^ 

...      .   ,  /  ,,  \  Tangents     from     a 

line  intersects  a }-  "  <  -   ,  ^    .1 

\  )      point  to  the  curve, 

curve  ;  I 


rASCAL'S  THEOREM.  341 

313.  Pascal's  Theorem.  If  a  hexagon  he  inscribed  in 
a  conic,  the  three  junction-points  of  its  three  pairs  of  opposite 
sides  lie  in  a  straight  line. 

Eemark.  By  a  polygon  inscribed  in  a  curve  is  meant  any 
chain  of  straight  lines,  returning  into  itself,  whose  consecutive 
junction-points  all  lie  on  the  curve.  A  polygoii  of  n  sides 
may  be  formed  by  taking  any  n  points  and  joining  them  con- 
secutively in  any  order  whatever. 


a  X  a'  a' 

Proof.  Let  1  2  3  4  5  6  be  the  inscribed  hexagon,  of  which 
the  opposite  sides  are 

12  and  45; 
23  and  5  6; 
3  4        and         6 1. 

Select  three  alternate  vertices,  as  2,  4  and  6,  and  consider 
the  pencils 

2-1345         and         6-134  5. 

Because  these  pencils  are  formed  by  joining  the  four  fixed 
points  1,  3,  4,  5  on  the  conic  to  the  points  2  and  6  respective- 
ly, their  anharmonic  ratios  are  equal  (§  311). 

Let  a,  h,  c,  d  be  the  row  of  points  in  which  the  pencil 
2-13  45  intersects  the  line  45.     We  shall  then  have 

{a,  h,  c,  d)  =  Anh.  ratio  (2-1345).  (§  296) 

Lei;  a',  ¥,  c,  d'  be  the  row  of  points  in  which  the  pencil 
6-1345  intersects  the  line  3  4.     We  have 

(a',  l\  c,  d')  =  Anh.  ratio  (6-13  45). 
Hence 

(a,  hf  c,  d)  =  {a',  h',  c,  d'). 


342  MODERN  GEOMETRY. 

These  two  rows  have  the  point  c  common  to  both,  and  this 
point  occupies  the  same  position  in  the  two  ratios.  There- 
fore the  three  lines 

aa',  hi',  dd' 
meet  in  a  point  (§  298). 

But  a  and  a'  are  the  respective  junction-points  (1  2  and  4  5) 
and  (6  1  and  3  4),  while  hb' =  ^d  and  dd'  =  5  6. 

Hence  the  opposite  sides  23  and  5  6  intersect  on  the  line 
aa'  which  joins  the  junction-points  of  the  two  other  pairs  of 
opposite  sides.     Q.  E.  D. 

314.  Correlative  of  Pascal's  Theorem:  Brian- 
chon's  Theorem.  The  three  lines  joining  the  opposite  ver- 
tices of  a  hexagon  cirmimscrihed  about  a  conic  meet  in  a  point. 

Proof.     Let  the  sides  taken  in  order  be  1,  2,  3,  4,  5,  6. 

Consider  the  two  rows  of  points  a,  Z>,  c,  d  and  a',  b',  c' ,  d' 
in  which  the  sides  1,  3,  4  and  5  intersect  2  and  6. 


Because  the  tangents  2  and  6  are  each  intersected  by  the 
four  tangents  1,  3,  4,  5,  we  have 

(a,  b,c,d)  =  (a',  b',  c\  d').  (§ 310) 

Consider  the  pencil  A-a,b,c,d.     We  have 

Anh.  ratio  {A-a,b,c,d)  =  (a,  b,  c,  d).  (§  296) 


J. i  _  j_ 


u 

K'p        'P,'P' 


_      ^^ 


Z' 


"1  =  1^^  S^  0^ 


X^-/^ ^^  -Lax-  '^^y. -f-  ^%^  i>'^'-r '^^  o 


=z  O 


y^ 


i 


TRILINEAR  CO-ORDINATES.  343 

Also,  for  the  same  reason 

Anh.  ratio  {A' -a' ,b\c' ^cV)  =  (a',  b\  c',  cV). 

Hence 

Anil,  ratio  (A-a,b,c,d)  =  Anli.  ratio  (-!'«',  ^',  c',  cV). 

Now  these  two  pencils  have  the  line  Ac  =  A'c'  common, 
and  occupying  the  same  position  in  the  two  ratios.  Hence 
the  three  remaining  lines  intersect  in  three  points  in  a  right 
line  (§298,  Cor.);  that  is,  the  three  points 

(e)  where  diagonal  Act  crosses  diagonal  A' a'-, 

(b)  where  line  Ab  crosses  A'y; 

{d')  where  line  Ad  crosses  A'd\ 
are  in  a  right  line,  which  proves  the  theorem. 

Trilinear  Co-ordinates. 

315.  In  the  method  of  trilinear  co-ordinates  the  posi- 
tion of  a  point  is  defined  by  its  relation  to  the  three  sides  of 
a  general  triangle.  Distances  from  each  side  are  considered 
positive  when  measured  in  the  direction  of  the  opposite  ver- 
tex; negative  in  the  opposite  case. 

Theorem.  If  a  general  triangle  be  given,  the  itosition  of 
a  point  is  completely  determined  tvhen  the  mutual  ratios  of 
its  distances  from  the  three  sides  of  the  triangle  are  given. 

Proof.     Let  it  be  given  that  ^fsA. 

the  distances  of  a  point  P  from 

the  sides  AB  and  ^C  of  a  tri-  cX  /  V 

angle  are  in  the  ratio  m  :  n. 

If  we  draw  a  line  Ax  divid- 
ing the  angle  BA  C  in  the  sine-  B^^^ — -^ ^  C 

ratio ,  every  point  of  this  line  will  fulfil  the  given  con- 
dition (§281). 

If  it  be  also  given  that  the  distances  of  the  point  P  from 
the  sides  ^Cand  BA  are  in  the  ratio  p  ;  m,  then  the  point 
P  must  also  lie  on  the  line  By  dividing  the  angle  B  in  the 

sine-ratio  —  — . 
m 


344  MODERN  GEOMETRY. 

Hence,  if  both  ratios  be  given,  the  only  point  which  will 
satisfy  them  is  the  junction-point  of  the  lines  Ax  and  By, 
which  is  therefore  the  required  point  P.     Q.  E.  D. 

Method  of  Expressing  the  Ratios  of  Distances.  The 
mutual  ratios  of  the  distances  of  a  point  P  from  the  three 
sides  of  a  triangle  are  most  conveniently  expressed  by  three 
numbers  proportional  to  these  distances.     Let  us  i3ut 

6/j,  d^,  6Z3,  the  distances  of  P  from  the  three  sides  of  the 
triangle; 

x^,  .^2,  x^,  any  three  numbers  proportional  to  d^,  d^,  d^. 
We  then  have 

(1) 


x^  :  X,  :  x^^d,\  d. 

'•d,, 

d: 

_  5..         d,  _x,^ 
x^ '         d,       x^ ' 

d,_  _  d,_  _  d. 

d^       X, 

also,  -'-  =  -"-  =  -^.  (3) 

^1       ^2       ^3 

If  we  put  p  for  the  common  value  of  the  three  fractions 
(3),  we  have 

d^  =  fix^\  J 

d^  ^  px^'\  (4) 

The  sets  of  equations  (1),  (2),  (3)  and  (4)  are  so  many 
different  ways  of  expressing  the  fundamental  fact  of  the  pro- 
portionality of  the  numbers  x^,  x^  and  x^  to  the  distances  d^, 
d^  "and  dy 

316,  Relation  hetween  the  Distances.  The  position  of 
the  point  is  completely  determined  when  its  actual  distances 
from  any  two  sides  of  the  triangle  are  given.  Hence  from  two 
distances  the  third  may  be  found,  which  shows  that  there  is 
some  equation  between  the  distances.     If  we  put 

a,  h,  c,  the  lengths  of  the  three  sides  of  the  fundamental 
triangle; 

A,  the  area  of  the  triangle, 
the  equation  in  question  is 

ad^  +  hd^  -f  cd^  =  2A.  (5) 


TRILINEAR  CO-ORDINATES.  845 

This  equation  is  readily  proved  by  drawing  lines  from  the 
point  to  the  three  vertices  and  equating  the  algebraic  sum  of 
the  areas  of  the  three  triangles  thus  formed  to  that  of  tlie 
original  triangle. 

When  .Tj,  x^  and  x^  are  given,  this  last  equation  with  the 
three  equations  (4)  suffice  for  the  determination  of  d^y  d^,  d^ 
and  p,  and  therefore  for  the  position  of  the  point.  In  fact,  by 
substituting  (4)  in  (5),  we  have  at  once 

p{ax^  +  hx^  +  cx^)  =  2  A, 

from  which  p  is  found.     Then  from  (4)  we  have  the  values  of 
^j,  d^  and  d^. 

31 T.  Multiplication  hy  Constant  Factors.  We  may,  in- 
stead of  taking  x^,  x^  and  x^  proportional  to  d^,  d^  and  d^^  take 
them  proportional  to  the  products  obtained  by  multiplying 
each  distance  d  by  any  arbitrary  but  constant  factor.  If  we 
take  /<!,  //j  and  yUg  for  such  factors,  we  shall  then  have 

x^:x^\x^=  }A^d^  :  }A^d^  :  fA^d^, 
or  fA^d,  =  px^] 


}^A  =  P^.',  \  (6) 

The  constant  factors  /j^,  //^  and  /j^  being  supposed  given, 
the  equations  (5)  and  (6)  suffice  for  the  determination  of 
d^,  d^_,  d^  and  p. 

318.  Definition  of  Trilinear  Co-ordinates.  The  tri- 
linear  co-ordinates  of  a  point  are  three  numbers  propor- 
tional to  the  distances  of  the  point  from  the  three  sides  of  a 
triangle,  each  distance  being,  if  we  choose,  first  multiplied  by 
any  fixed  factor. 

The  triangle  from  whose  sides  the  distances  are  measured 
is  called  the  fundamental  triangle. 

Corollaries  from  the  Definition  : 

I.  If  the  trilinear  co-ordinates  of  a  point  be  all  multiplied 
lij  the  same  factor,  the  position  of  the  point  which  they  repre- 
sent will  7iot  le  altered. 

For  the  position  of  the  point  depends  only  on  the  mutual 


346  MODERN  OEOMETRT. 

ratios  of  its  trilinear  co-ordinates,  which  remain  unaltered  by 
such  multiph'cation. 

11.  The  points  (1),  (2)  and  (3)  wliose  respective  co-ordi- 
nates are : 

Point  (1),  {x^,  0,  0), 

Point  (2),  (0,  x^,  0), 

Point  (3),  (0,  0,  x^), 

are  the  three  vertices  of  the  fundamental  triangle,  no  matter 
what  the  absolute  values  of  x^,  x^,  and  x^. 

Note.  The  introduction  of  the  factors  //  being  a  mere  matter  of 
convenience,  the  student  may  ordinarily  leave  them  out  of  consideration, 
which  is  the  same  as  to  suppose  them  unity.  Their  introduction 
amounts  to  supposing  that  the  distances  from  the  different  sides  of  the 
triangle  may  be  expressed  in  three  different  units  of  length  without 
destroying  the  truth  of  our  conclusions. 

EXERCISES. 

Prove : 

1.  The  distances  of  a  point  from  the  three  sides  of  the 
fundamental  triangle  cannot  all  be  negative. 

2.  Assuming  the  factors  /^j,  fx^  and  fx^  to  all  have  the  same 
sign,  every  point  whose  co-ordinates  are  all  positive  or  all 
negative  lies  within  the  fundamental  triangle. 

3.  If  the  factors  fj.  are  all  positive  and  the  trilinear  co- 
ordinates of  a  point  all  negative,  the  value  of  p  must  be 
negative. 

4.  The  lines  from  the  point  {x^,  x^,  x^)  to  the  three  vertices 
of  the  triangle  divide  the  internal  angles  in  the  respective  sine- 
ratios 

/^,<  M,^,'  ^^< 

319.  Eelation  of  Trilinear  and  Rectangular  Co-ordinates. 
Let  us  suppose  the  three  sides  of  the  fundamental  triangle  to 
be  given  by  their  general  equations  in  rectangular  co-ordinates, 
as  follows: 

Side  1,  ax    -\-  hy    -\-  c    =0; 

Side  2,  a'x  +  h'y  -f  c'  =  0;  J-  (7) 

Side  3,  a"x  +  l)"y  -|-  c"  =  0. 


TRILINEAR    CO-ORDINATES.  347 

Then,  by  §§  41,  54,  if  x  and  y  be  the  rectangular  co-ordinates 
of  any  point  whatever,  the  expression 

ax  -\-  by  -[-  c  (a) 

will  represent  the  distance  of  that  point  from  the  side  (1)  of 
the  triangle,  multiplied  by  the  factor  Va^  +  Jf,  Since,  by 
multiplying  the  equation  of  side  (1)  by  an  appropriate 
factor,  we  can  give  the  quantity  Va^  +  H^  any  value  we  please, 
we  can  make  it  equal  to  //j.  We  shall  then  have,  when  x  and 
y  are  the  rectangular  co-ordinates  of  a  point  distant  d^  from 
the  side  (1), 

//^fZj  =  ax  -\-  hy  -\-  c. 

Thus  the  equations  (6)  of  §  317  may  be  replaced  by 

ax     -{-hi/     -\-  c     =  px^;  j 

a'x   +  yy   +  c'    =  px,;  [  (8) 

a''x  +  b"y  +  c"  =  px.^.  ) 

These  equations  determine  p,  x  and  y  when  x^,  x^  and  x^  are 
given.     The  values  of  x  and  y  thus  obtained  from  them  are 


_  (yc''  -  h"c')x^  +  {h"c  -  lc")x^  +  {he'  -  h'c)x^ 
^  ~  (a'y  -  a"b')x^  +  (a''h  -  ah'')x^  +  {ah'  -  a'h)xj 

-  («'V  -  a'c")x^  +  {(^io"  -  cl"c)x„_  +  {a'c  -  ac')x^^ 
^  ~  {a'h"  -  a"b')x,  +  {a"h  -  ab")x,  +  {ah'  -  a'h)x^' 


(9) 


which  are  the  expressions  for  the  rectangular  co-ordinates  of 
a  point  {x.  y)  in  terms  of  its  trilinear  co-ordinates. 

320,  Equation  of  a  Straight  Line  in  Trilinear  Co-ordi- 
nates. 

Theorem.  If  the  trilinear  co-ordinates  of  a  point  are  re- 
quired to  satisfy  a  linear  equation,  the  locus  of  the  point  will 
he  a  straight  line. 

Proof,     Let 

P,X^    +  P,^;    +  i?3^3    =    ^ 

be  the  linear  equation  which  the  co-ordinates  must  satisfy. 
If  we  substitute  ioxx^,  remand  x^  their  expressions  (8)  in  terms 
of  Cartesian  co-ordinates,  we  readily  see  that  the  equation 


348  MODERN  GEOMETRY, 

will  be  of  the  first  degree  in  x  and  y.  It  is  therefore  the 
equation  of  a  straight  line. 

321.  Homogeneous  Character  of  Equations.  In  order 
that  any  equation  in  trilinear  co-ordinates  may  represent  a 
locus,  the  equation  must  be  homogeneous  in  terms  of  such 
co-ordinates.  For,  by  definition,  the  position  of  a  point  re- 
mains unaltered  when  its  three  co-ordinates  are  all  multi2)lied 
by  any  arbitrary  common  factor  p.  When  we  take  a  homo- 
geneous equation  of  the  ?ith  degree  in  x^,  x^  and  x^,  such,  for 
example,  as  (when  n  =  2) 

ax^x^  +  Ix^  —  0, 

and  multiply  x^,  x^  and  x^  by  p,  the  result  is  the  same  as  if 
we  multiplied  each  member  of  the  equation  by  p".  Hence 
the  relation  between  x^^  x^  and  x^  expressed  by  the  equation 
remains  unaltered.     But  if  we  take  such  an  equation  as 

ax^x^  +  Ix^  -f  c  =  0, 

and  multiply  x^,  x^  and  x^  by  c,  the  result  is 

ap^x^x^  +  hpx^  -\-  c  =  0, 

an  equation  which  expresses  a  different  relation  from  the 
other.  Hence  such  an  equation  cannot  represent  a  definite 
locus  so  long  as  the  trilinear  co-ordinates  are  used  to  corre- 
spond to  their  definition. 

Correlative  of  Trilinear  Co-ordinates. — 
Co-ordinates  of  a  Line. 

322.  The  principle  of  duality  is  applicable  to  all  the  pre- 
ceding propositions  which  express  position.  We  shall  there- 
fore change  these  propositions  into  their  correlatives. 

Theorem.  If  three  fixed  points  not  in  the  same  straight 
line  le  given,  the  position  of  a  line  is  completely  determined 
when  the  mutual  ratios  of  its  distances  from  these  points  are 
given. 

Proof.     Let  the  three  fixed  points  be  A,  B  and  C. 

Let  it  be  given  that  the  ratio  of  the  distances  of  a  line 
from  the  points  A  and  C  is  m  :  w. 


TRILINEAR  CO-ORDINATES.  349 

Let  L  be  any  Hue  i'ullilling  this  condition,  and  let  y  be 
the  point  in  which  it  cuts  the  line  AC.     Also  let  ^la'and  6V 


C  B 


be  tlie  perpendiculars  from  A  and  G  upon  the  line  L»     We 
then  have,  by  hypothesis, 

Aa'  :  Cc'  :=^m  :  n. 

Hence,  by  simil?,r  triangles. 

Ay  \  Cy  =.  m:  n. 

This  last  relation  completely  fixes  the  position  of  the 
point  y  (§  279),  which  therefore  remains  the  same  for  all  lines 
which  satisfy  the  given  condition.     That  is. 

Every  line  fulfilling  the  condition  that  its  distances  from 
two  fixed  points,  A  and  C,  shall  he  in  the  ratio  m  :  n,  j^asses 
through  that i^oint  luhich  divides  the  junction-line  AC  in  the 
ratio  m  :  n. 

Let  it  also  be  given  that  the  distances  of  the  line  from  the 
points  C  and  B  shall  be  in  the  ratio  n  :  p.  It  must  then  pass 
through  a  point  X  which  divides  the  junction-line  CB  in  the 
ratio  n  -.p. 

If  now  it  be  required  that  the  line  shall  satisfy  both  con- 
ditions, it  must  pass  through  both  the  points  Xand  y,  and  is 
therefore  completely  fixed. 

When  both  these  conditions  are  satisfied,  the  distances  of 
the  line  from  BA  will  be  in  the  ratio  p  :  m,  and  it  will  inter- 
sect AB  in  some  point,  dividing  AB  in  the  ratio  jt?  :  m. 

The  three  points  then  satisfy  the  proposition  of  (§  287), 
because 

m    n     P  _  -, 
n  '  p  '  m  ~ 


350  MODERN  OEOMETRT. 

Proceeding  as  in  the  case  of  the  point,  if  we  put 
i)j,  D^,  Z^g,  the  distances  of  the  line  from  the  three  points; 
A  J,  Ag,  Ag,  constant  factors, 
we  may  express  the  mutual  ratios  of  the  distances  by  three 
quantities,  u^,  u^  and  tc^,  proportional  to  them.    We  then  have 

W|  :  tc,  :  u,  =  \^D^  :  XJ)^  :  X^D,; 

A,A  ^  ^A  ^  ^sA  -  ^. 
w,  ti^  u,    -^' 

X^D^  =  Gu^, 

Thus  the  position  of  the  line  is  completely  fixed  by  the 
three  quantities  Wj,  u^  and  u^,  which  are  therefore  called  co- 
ordinates of  the  line.     We  therefore  have  the  definition: 

The  trilinear  co-ordinates  of  a  line  are  three  numbers 
proportional  to  its  distances  from  three  fixed  points,  each 
distance  being  first  multiplied  by  any  fixed  factor. 

The  junction-lines  of  the  three  points  form  the  funda- 
mental triangle  of  reference. 

323.  There  are  therefore  two  ways  of  defining  the  posi- 
tion of  a  line,  namely: 

1.  By  the  equation  of  the  line. 

2.  By  the  value  of  its  co-ordinates. 

To  investigate  the  relation  of  these  two  ways  let  us  con- 
sider the  problem :  What  are  the  co-ordinates  of  the  line  whose 
equation  is 

rnx^  +  nx^  +  px^  =  0?  (a) 

Let  us  suppose  the  indices  ^\G 

1,  2  and  3  to  refer  to  the  sides  2./^^    \ 

BCf  CA  and  AB  respectively.         ^^  \ 

Let  us  first  find  the  point  x  in^^/_ \ ^ 

which  the  line  (a)  intersects  ^  ^ 

AB,     To  do  this  we  put  x^  =  0,  which  gives 

^1   _        ^  . 

X„    ~  7?l* 


TRILINEAR  CO-ORDINATES.  351 

and  by  substituting  for  a;,  and  x^  tlieir  expressions  (6)  in  terms 
of  the  distances, 

d,  _  _  /VL 

d^  M.m ' 

Since  d^  and  d^  are  the  distances  of  x  from  the  sides  CA  and 

CB,  —  --  is  the  sine-ratio  in  which  the  line  Cx  divides  the 

angle  C  (§  281).    To  define  the  point  x  by  the  distance-ratio  in 
which  it  cuts  AB,  we  have  the  equations 

Bx       sin  BCx 


Cx         sinB   ' 

Ax      sin  A  Cx 

Cx  ~~     sin  ^    * 

Hence 

• 

Bx      sin  BCx 

Ax  ~  sin  A  Cx 

sin  A  _        d^  sin  A 
'  sin  B  ~       d^  sin  B 

_  /j^n  sin  ^ 
"  //,7?z  sin  B' 

(b) 

which  determines  the  point  of  intersection,  x. 

By  what  has  been  shown,  this  distance-ratio  is  the  ratio  of 
the  two  co-ordinates  u^  and  u^  of  the  line,  multiplied  by  a 
factor.  In  fact,  putting  D^  and  D^  for  the  distances  of  the 
line  from  A  and  B  respectively,  we  have 


Bx 

A 

X 

,u^ 

Ax~ 

"A 

~  X 

,u. 

Comparing 

this  with  the  equati 

on  (b),  we 

)  have, 

for  the  ratio 

of  the  two  line-coordinates. 

u,_ 

X^f^^n  sin 

A 

n 

K^. 

!  sin 

iA 

^1 

A,yUjWsin 

B  ■ 

m 

'Kf^. 

sin 

B' 

In  the  same 

1  way, 

W3 

Kf^sP  sin 
A^yw^^^  sin 

B 

C~ 

■I 
n 

sin 

sin 

B 

(.0) 

Mr_ 

A,/i,msin 

C 

m 

\f^. 

sin 

C 

M, 

\pi^p  sin 

A  ~ 

P 

■  Kf^. 

sin 

A' 

Now,  it  will  be  remembered  that  the  constant  factors  X 
and  p.  which  multiply  the  distances  between  the  points  and 


353  MODERN  OEOMETRY. 

lines  have  been  left  entirely  arbitrary.  Can  we  not  so  deter- 
mine them  that  the  last  fractions  in  the  third  members  of  the 
above  equations  shall  be  unity?  This  will  require  the  factors 
to  fulfil  the  conditions 

Ti^fx^  sin  B  =  X^fJi^  sin  ^;  j 

A^yw,  sin  C  —  Ag/Zg  sin  i^;  >  {d) 

X^l^t^  sin  A  =  Aj//,  sin  C.  ) 

These  three  equations  are  really  equivalent  to  but  two,  be- 
cause any  one  can  be  deduced  from  the  other  two  by  eliminat- 
ing the  common  angle.  If  we  suppose  the  values  of  /i  to  be 
given,  we  can  determine  the  mutual  ratios  of  the  A's  by  the 
equations 

A^  _  /fg  sin  ^  ^ 
Ag  ~~  /*,  sin  C^ 

K  _  /^3  sin  B 
Ag  ~  jLi^  sin  C 

Hence  the  required  condition  can  always  be  satisfied,  and  we 
shall  always  suppose  it  satisfied. 

The  equations  (c)  can  then  be  satisfied  by  putting 

u^  =  m  X  any  factor; 

u^  =  n  X  the  same  factor; 

Wg  —  p  X  the  same  factor. 
Hence: 

Theorem.  If  the  co-ordinates  x^,  x^,  x^  of  a  point  are 
considered  as  variaUes  required  to  satisfy  the  equation 

mx^  4-  nx^  4-  px^  =  0, 

the  point  will  always  lie  on  the  line  whose  co-ordinates  are  the 
constants  m,  n  and  p,  or  their  multiples. 

324.  Equation  of  a  Point.  We  have  the  following 
theorem,  the  correlative  of  the  preceding  one. 

Theorem.  All  lines  whose  co-ordinates  ti^,  u^  and  u^ 
satisfy  a  linear  equation 

mu^  +  nu^  +  pu^  =  0 


TRILINEAR   CO-ORDINATES.  353 

pass  through  a  point,  namely,  the  point  whose  co-ordinates  are 
m,  n  and  p. 

This  theorem  follows  immediately  from  the  preceding  one, 
because  when  a  point  lies  on  a  certain  line,  the  line  passes 
through  that  point.     Putting  both  equations  in  the  form 

u^x^  -f  ii^x^  +  %(,^x^  ^  0, 

the  theorem  of  §  323  asserts  that  whenever  this  equation  is 
satisfied  the  point  {x^,  x^,  xj  lies  upon  the  line  (w,,  u^,  2iJ. 
Changing  the  form  but  not  the  essence  of  the  conclusion,  we 
have  the  theorem  that  whenever  this  equation  is  satisfied  the 
line  (n^,  w,,  ti^)  passes  through  the  point  (rr,,  x^,  x^).  This 
result,  being  true  for  all  values  of  the  six  quantities  which 
satisfy  the  equation,  will  remain  true  when  we  suppose  x^,  x^ 
and  x^  to  remain  constant  and  w,,  u^  and  Wg  to  vary;  that  is, 
the  varying  line  {u^,  u^,  u^)  will  then  constantly  pass  through 
the  fixed  point  (a:,,  x^,  x^). 

325.  The  preceding  conclusions  may  be  summed  up  as 
follows: 

I.  Ifu^,  ic^  and  u^  are  line  coordinates,  and  x^,  x^  and  x^ 
are  point-coordinates,  then,  so  long  as  these  co-ordinates  are 
unrestricted,  they  may  represent  ayiy  line  and  ayiy  point  what- 
ever. 

II.  If  it  he  required  that  the  line  shall  pass  through  the 
point  and  the  point  lie  on  the  line,  the  co-ordi7iates  must  satisfy 
the  condition 

u^x^  +  u^x^  +  u,x^  =  0. 

III.  If,  in  this  equation,  we  suppose  the  x^s  to  remain 
fixed  while  the  u^s  vary,  the  lines  represented  hy  the  ti/s  tuill 
all  pass  through  the  fixed  point  represented  hy  the  x's. 

IV.  If  we  suppose  the  x's  to  vary  while  the  u's  remain  con- 
stant, the  points  represented  hy  the  x^s  will  all  lie  on  the  fixed 
line  represented  hy  the  it's. 

336.  For  brevity  of  writing  we  may  use  a  single  letter  to 
represent  the  combination  of  three  co-ordinates  of  a  point  or 
line.  Then  the  expression  **the  point  {p)"  will  mean  the 
point  whose  co-ordinates  are  />„  p^  and  p^. 


354 


MODERN  OEOMETRT. 


Theorem.     If  {x)  and  (y)  are  a?iy  two  points^  and  if, 
with  any  factor  A,  we  form  the  quantities 

^1  —  ^1  4"  ^Vi^  ) 

z^  =  x^-\-  Xy„  V  {a) 

2^3  =  ^3  +  ^Vz^  ' 
the  point  (z)  luill  lie  on  the  li^ie  joining  the  points  (:f)  and  {y). 
Proof.     Let  (p)  be  the  line  joining  the  points  (a:)' and  (y). 
Because  the  line  {p)  passes  through  the  point  {x),  we  have 

Because  {p)  passes  through  the  point  {y),  we  also  have 

p,y^  +  P.y.  +  ^^3^3  =  0. 
Multiplying  this  equation  by  X  and  adding  it  to  the  other, 

pM.  +  ^y.)  +  pX^^.  +  ^.)  +  pA^.  +  ^^3)  =  0, 

or  ;?,2j  +  ^2^2  +  i^3^»  =  ^-  .       ■ 

Therefore  the  point  {z)  is  on  the  line  (p).     Q.  E.  D. 

Cor.     Any  point  (2;)  whose  co-ordinates  satisfy  the  con- 
ditions 

z^  =  Xx^  +  /uy„ 

z,  =  Xx^  +  My^, 

z,   =   ^x,  +  >"2/3» 

lies  on  the  same  straight  line  with  the  points  (x)  and  (//),  what- 
ever be  the  factors  X  and  yw.* 

327.  Problem.     Having  the  four  2')oints  in  a  right  line, 
{x),         (y),         {x-{-Xy),         {x-^rX'y), 

it  is  required  to  determi7ie  their  a7iharrnonic  ratio. 

We  must  first,  instead 
of   the   trilinear   co-ordi-  v^ 

nates,  take  the  actual  re- 
duced   distances    of    the 
points  from  the  sides  of  A" 
the  triangle.     Let  us  then  suppose 

*  These  coefficients  A  and  f.i  must,  of  course,  not  be  confounded  with 
the  factors  used  in  g§  317,  322. 


TRILINEAR  CO-ORDINATES.  355 


AB,  any  side  of  tlie  fundamental  triangle; 

z  E  the 

point  {x-\-\y)', 

ax,  by. 

cz  ^  p,  q 

and  r,  the  distances  of  x,  y  and  z  from 

AB. 

We  then  have  for 

the  distance-ratio  of  z,  with  respect  to 

X  and  y, 

xz       r  —  p 
yz       r  -  q      "^ 

This  gives 

j.-J>  -  ^9 

1-M 

We  have  three  equations  of  this  form,  corresponding  to 
the  three  sides  of  the  fundamental  triangle,  in  which  p,  q 
and  r  have  the  respective  indices  1,  2  and  3.  We  may  write 
them: 

(1  -  M)r^  =p,  -  MQr; 

(1  ~  M)r,  =  p,  -  Mq,; 

{l  -  M}r,  =  p,  -  Mq^' 

The  trilinear  co-ordinates  a;,,  x^  and  x^,  proportional  to 

jOj,  p^  and  ^3,  are  formed  by  dividing  these  last  quantities  by 

a  common  factor  =  p.     Let  <j  be  the  common  factor  for  q. 

Then  these  equations  become,  by  substitution  and  reduction. 


'-^'■ 

=  ^,  • 

'^'■ 

=  ^,  • 

. 

'-^" 

=    ^3     ■ 

These  equations 

become  identical 

with  {a)  by  putting 

1- 

z  =  

P 

''         P 

=  K 

or 

"-^ 

But  yu  is  the  distance-ratio  of  the  point  {z)  =  (a:  -f  ^v)  with 
respect  to  the  points  {x)  and  (?/).     Hence, 

When  from  two  poijits,  (x)  and  (y),  we  form  the  third, 
(x  -\-  \y),  in  the  same  straight  line,  the  distance-ratio  of  the 


356  MODERN  GEOMETRY. 

third  tuith  respect  to  the  other  two  is  equal  to  X  multijMed  hj 

a  factor,  —  — ,  depending  ui^on  the  absolute  values  of  the  tri- 

lincnr  co-ordinates. 

Since,  from  the  nature  of  trilinear  co-ordinates,  this  factor 
remains  indeterminate,  the  distance-ratio  also  remains  inde- 
terminate. But  if  we  also  take  the  distance-ratio  of  a  fourth 
point,  (x  +  ^'y),  and  then  form  the  anharmonic  ratio,  this 
factor  will  divide  out,  and  we  shall  have 

Anh.  ratio  =  jj. 

Hence  the  anharmonic  ratio   of  the   four   points   (x),  (y), 

(x  +  Xy),  {x  -\-  X'y)  is  -77,  which  solves  the  problem. 

Cor.  Harmonic  Points.  Since  four  harmonic  points  are 
such  whose  anharmonic  ratio  is  —  1,  we  must  then  have 
A'  =  —  /I.  We  therefore  conclude  that  if  we  have  any  four 
points  capable  of  being  expressed  in  the  form 

W.       {y),       (^  +  '^y)y        {^  -  ^)y 

the  last  pair  will  divide  harmonically  the  segment  contained 
by  the  first  pair. 


OOUESE  OF  EEADING  IN  GEOMETET. 


The  following  classified  list  of  books  is  prepared  for  the 
use  of  students  who  wish  to  continue  the  study  of  the  subject: 

I.  MODERN  SYNTHETIC  GEOMETRY. 

Chasles,  Traite  de  Geomelrie  SupSrieure  (546  pages  8vo),  is  noted  for 

its  elegance  of  treatment.     It  is  principally  confined  to  tlie  geometry 

of  lines  and  circles.     The  subject  is  continued  in 
Chasles.  Traite  des  Sections  Coniques,  Premiere  Partie^{no  second  part 

publislied), 
TowNSEND,  Modern   Geometry  of  the   Point,  Line  and  Circle  (2  vols. ; 

Dublin,  1863),  covers  ground  similar  to  the  first  work  of  Chasles, 

but  is  more  elementary. 
Steener,    Vorlesungen  iXber  Synthetische  Geometrie,  is  a  very  extended 

treatise,  but  lacks  the  clear  presentation  of  Chasles. 

II.    PLANE  ANALYTIC  GEOMETRY. 

Salmon's  Conic  Sections  and  his  Higher  Plane  Curves  treat  the  subject 
with  the  clearness  and  simplicity  which  characterize  the  works  of 
that  author. 

Clebsch,  Vorlesungen  ilher  Geometrie,  of  which  there  is  a  French 
translation,  is  the  most  complete  single  treatise  on  the  higher 
branches  of  modern  geometry  now  at  the  command  of  the  student. 

III.     ANALYTIC  GEOMETRY  OF  THREE  DIMENSIONS. 

Salmon,  Analytic  Geometry  of  Three  Dimensions,  is  the  most  extended 
treatise  in  English.  It  presupposes  a  knowledge  of  the  elements 
of  modern  algebra,  such  as  can  readily  be  derived  from  his  treatise 
on  that  subject. 

Frost,  Solid  Geometry,  is  less  extended  than  Salmon's  treatise,  but 
written  more  in  the  style  of  a  text-book. 

Aldis,  Solid  Geometry  (223  pages  12mo),  is  an  excellent  little  elementary 
text-book,  with  numerous  exercises. 

Hesse,  Voi'lesungen  iiber  Analytische  Geormtrie  des  Raumes,  is  a  Ger- 
man work  of  450  pages,  noted  for  its  elegance  of  treatment. 


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